HYDRAULICS 


BY 


F.   C.   LEA, 

B.Sc.  (LONDON  ENGINEERING), 


SENIOR   WHITWORTH    SCHOLAR;    ASSOC.R.  COL.  SO. ;    A.M.INST.  C.  E.; 

TELFORD   PRIZEMAN  ;    LECTURER   IN   APPLIED   MECHANICS   AND   ENGINEERING  DESIGN 
IN   THE    CITY   AND    GUILDS    OF   LONDON    CENTRAL   TECHNICAL   COLLEQ2. 


OF  THE 

UNIVERSITY 

OF 


NEW   YORK: 

LONGMANS,    GREEN   &   CO 
LONDON:    EDWARD   ARNOLD 

[All  Rights  reserved] 


'\  A**1^ 

OF  THE 

f    UNIVERSITY   ] 

OF 
C*L!F> 


PREFACE. 

WHEN  the  author  undertook  some  time  ago  to  write  this 
work,  it  was  under  the  impression,  which  impression  was 
shared  by  many  teachers,  that  a  book  was  required  by  Engineering 
students  dealing  with  the  subject  of  Hydraulics  in  a  wider  sense 
than  that  covered  by  existing  text  books.  In  addition  the  author 
felt  that  though  several  excellent  text  books  were  in  existence, 
the  large  amount  of  experimental  research  carried  out  during  the 
last  10  or  15  years,  very  little  of  which  has  been  done  in  this 
country,  on  the  subject  of  the  flow  of  water,  had  not  received  the 
attention  it  deserved.  The  great  developments  in  turbines  and 
centrifugal  pumps  also  merited  some  notice. 

An  attempt  has  been  made  to  embody  the  results  of  the  latest 
researches  in  the  book,  and  to  give  sufficient  details  to  indicate 
the  methods  used  in  obtaining  these  results,  especially  in  those 
cases  where  such  information  and  the  references  thereto,  are 
likely  to  prove  of  value  to  those  desirous  of  carrying  out  ex- 
periments on  the  flow  of  water. 

Perhaps  in  no  branch  of  Applied  Science  is  it  more  difficult 
to  co-ordinate  results  and  express  them  by  general  formulae  than 
in  Hydraulics.  Practical  Engineers  engaged  in  the  design  of 
water  channels  frequently  complain  of  the  large  differences  they 
obtain  in  the  calculated  dimensions  of  such  channels  by  using 
the  formulae  put  forward  by  different  authorities.  Before  any 
formula  can  be  used  with  assurance  it  is  necessary  to  have  some 
knowledge  of  the  data  used  in  determining  the  empirical  con- 
stants in  the  formula.  For  this  reason  a  little  attention  has  been 
given  to  the  historical  development  of  the  formulae  for  determining 
the  flow  in  pipes  and  channels,  and  some  particulars  of  the  data 
from  which  the  constants  were  determined  are  given.  In  this 
respect  the  logarithmic  analysis  of  experimental  data,  especially 
in  Chapter  YI,  together  with  the  plottings  of  Fig.  114  and  the 
references  to  experiments,  will  it  is  hoped  be  of  assistance  to 


IV  PREFACE 

engineers  in  enabling  them  to  choose  the  coefficients  suitable  to 
given  circumstances,  and  it  is  further  hoped  that  the  methods  of 
analysis  given  will  be  educational  and  useful  to  students,  and 
helpful  in  the  interpretation  of  experiments. 

The  chapter  on  the  flow  of  water  in  pipes  is  arranged  so  that  a 
student  who  reads  as  far  as  section  93  should  be  able  to  solve 
a  large  number  of  problems  on  flow  of  water  in  pipes,  without 
further  reading.  At  the  end  of  the  chapter  the  formulae  derived 
in  the  chapter  are  summarised,  and  various  kinds  of  practical 
problems  solved,  and  arithmetical  examples  worked  out.  In  the 
chapter  on  flow  in  channels  the  student  who  reads  to  section  119, 
and  then  sections  124  and  129  should  be  able  to  follow  the 
problems  at  the  end  of  the  chapter,  and  to  work  the  examples. 
Chapter  YIII  enables  the  student  who  is  desirous  of  studying 
the  elementary  theory  of  the  impact  of  water  on  vanes,  and  of 
turbines,  to  do  so  apart  from  the  details  of  turbines,  and  the  more 
practical  problems  that  arise  in  connection  with  their  design. 

The  principles  of  construction  of  the  various  types  of  turbines 
are  illustrated  in  Chapter  IX  by  diagrams  of  the  simpler  and 
older  types,  as  well  as  by  drawings  of  the  more  complicated 
modern  turbines.  The  drawings  have  been  made  to  scale,  and 
in  particular  cases  sufficient  dimensions  are  given  to  enable  the 
student  acquainted  with  the  principles  of  machine  design  to 
design  a  turbine.  The  author  believes  the  analysis  given  of  the 
form  of  the  vanes  for  mixed  flow  turbines  and  also  for  parallel 
flow  turbines  is  new. 

The  subject  of  centrifugal  pumps  is  treated  somewhat  fully, 
because  of  the  complaint  the  author  has  often  heard  of  the 
difficulty  engineers  and  students  have  in  determining  what  the 
performance  of  a  centrifugal  pump  is  likely  to  be  under  varying 
conditions.  The  method  of  analysis  of  the  losses  at  entrance  and 
exit  as  given  in  the  text,  the  author  believes,  is  due  to  Professor 
Unwin,  and  he  willingly  acknowledges  his  obligation  to  him. 
The  general  formula  given  in  article  237  is  believed  to  be  new, 
and  the  examples  given  of  its  application  in  sections  235,  etc., 
show  that  by  such  an  equation,  which  may  be  called  the  character- 
istic equation  for  the  pump,  the  performance  of  the  pump  under 
varying  conditions  can  be  approximately  determined. 

The  effects  of  inertia  forces  in  plunger  pumps  and  the  effect  of 
air  vessels  in  diminishing  these  forces  are  only  imperfectly  treated, 
as  no  attempt  is  made  to  deal  with  the  variations  of  pressure  in 
the  air  vessel.  Sufficient  attention  is  however  given  to  the 
subject  to  emphasise  the  importance  of  it,  and  it  is  probably 
treated  as  fully  as  is  desirable,  considered  from  a  practical 


PREFACE  v 

engineering  standpoint.  The  analysis  of  section  260,  although 
too  refined  for  practical  purposes,  is  of  value  to  the  student  in 
that,  neglecting  losses  which  cannot  very  well  be  determined,  it 
enables  him  to  realise  how  the  energy  given  as  velocity  head  to 
the  water  both  in  the  cylinder  and  in  the  suction  pipe  is  recovered 
before  the  end  of  the  stroke  is  reached.  The  examples  given  of 
"  Hydraulic  Machines  "  have  been  chosen  as  types,  and  no  attempt 
has  been  made  to  introduce  very  special  kinds  of  machines.  The 
author  has  had  a  wide  experience  of  this  class  of  machinery,  and 
he  thinks  the  examples .  illustrate  sufficiently  the  principles  and 
practice  of  the  design  of  such  machines. 

The  last  two  chapters  have  been  introduced  in  the  hope  that 
they  will  be  of  assistance  to  University  students,  and  to  candidates 
for  the  Institution  of  Civil  Engineers  examinations. 

Mr  Froude's  experiments,  on  the  frictional  resistance  of  boards 
moving  through  water,  are  considered  in  Chapter  XII  simply  in 
their  relationship  to  the  resistance  of  ships,  and  no  attempt  has 
been  made,  as  is  frequently  done,  to  use  them  to  determine  so- 
called  laws  of  fluid  friction  for  water  flowing  in  pipes  and 
channels. 

The  author  hardly  dares  to  hope  that  in  the  large  amount  of 
arithmetical  work  involved  in  the  exercises  given,  mistakes  will 
not  have  crept  in,  and  he  will  be  grateful  if  those  discovering 
mistakes  will  kindly  point  them  out. 

The  author  wishes  to  express  his  sincerest  thanks  to  his 
friend,  Mr  W.  A.  Taylor,  Wh.Sc.,  A.R.C.S.,  for  his  kindness  in 
reading  proofs,  and  for  many  valuable  suggestions,  and  also  to 
Mr  W.  Hewson,  B.Sc.,  who  has  kindly  read  through  some  of  the 
proofs. 

To  the  following  firms  the  author  is  under  great  obligation  for 
the  ready  way  in  which  they  acceded  to  his  request  for  information : 

Messrs  Escher,  Wyss  and  Co.  of  Zurich  for  drawings  of 
turbines  and  for  loan  of  block  of  turbine  filter. 

Messrs  Piccard,  Pictet  and  Co.  of  Geneva  for  drawings  of 
turbines. 

Messrs  Worthington  and  Co.  for  drawings  of  centrifugal 
pumps  and  for  loan  of  block. 

Messrs  Fielding  and  Platt  of  Gloucester  for  drawings  of 
accumulator. 

Messrs  Tangye  of  Birmingham  for  drawings  of  pumps. 

Messrs  Glenfield  and  Kennedy  of  Kilmarnock  for  drawings  of 
meter  and  for  loan  of  blocks. 

Messrs  G.  W.  Kent  of  London  for  description  and  loan  of 
blocks  of  Venturi  meter  recording  gear. 


V]  PREFACE 

Messrs  W.  and  L.  E.  Gurley  of  Troy,  N.Y.,  U.S.A.  for  loan 
of  block  of  current  meter. 

Messrs  Holden  and  Brooke  of  Manchester  for  drawing  of 
Leinert  meter. 

Messrs  W.  H.  Bailey  and  Co.  of  Manchester  for  drawing  of 
hydraulic  ram. 

Messrs  Armstrong,  Whitworth  and  Co.  for  drawings  of  crane 
valves. 

Messrs  Davy  of  Sheffield  for  loan  of  block  of  forging  press. 


F.  C.  LEA. 


CENTRAL  TECHNICAL  COLLEGE, 
November,  1907. 


CONTENTS. 
CHAPTER  I. 

FLUIDS   AT   REST. 

Introduction.  Fluids  and  their  properties.  Compressible  and  incom- 
pressible fluids.  Density  and  specific  gravity.  Hydrostatics.  Intensity 
of  pressure.  The  pressure  at  a  point  in  a  fluid  is  the  same  in  all  directions. 
The  pressure  on  any  horizontal  plane  in  a  fluid  must  be  constant.  Fluids 
at  rest  with  free  surface  horizontal.  Pressure  measured  in  feet  of  water. 
Pressure  head.  Piezometer  tubes.  The  barometer.  The  differential  gauge. 
Transmission  of  fluid  pressure.  Total  or  whole  pressure.  Centre  of 
pressure.  Diagram  of  pressure  on  a  plane  area.  Examples  .  Page  1 


CHAPTER  II. 

FLOATING  BODIES. 

Conditions  of  equilibrium.  Principle  of  Archimedes.  Centre  of 
buoyancy.  Condition  of  stability  of  equilibrium.  Small  displacements. 
Metacentre.  Stability  of  rectangular  pontoon.  Stability  of  floating  vessel 
containing  water.  Stability  of  floating  body  wholly  immersed  in  water. 
Floating  docks.  Stability  of  floating  dock.  Examples  .  .  Page  21 


CHAPTER  III. 

FLUIDS   IN   MOTION. 

Steady  motion.  Stream  line  motion.  Definitions  relating  to  flow  of 
water.  Energy  per  pound  of  water  passing  any  section  in  a  stream  line. 
Bernouilli's  theorem.  Venturi  meter.  Steering  of  canal  boats.  Extension 
of  Bernouilli's  theorem.  Examples Page  37 


viii  CONTENTS 

CHAPTER  IV. 

FLOW  OF   WATER  THROUGH   ORIFICES   AND   OVER   WEIRS. 

Velocity  of  discharge  from  an  orifice.  Coefficient  of  contraction  for 
sharp-edged  orifice.  Coefficient  of  velocity  for  sharp-edged  orifice.  Bazin's 
experiments  on  a  sharp-edged  orifice.  Distribution  of  velocity  in  the  plane 
of  the  orifice.  Pressure  in  the  plane  of  the  orifice.  Coefficient  of  discharge. 
Effect  of  suppressed  contraction  on  the  coefficient  of  discharge.  The  form 
of  the  jet  from  sharp-edged  orifices.  Large  orifices.  Drowned  orifices. 
Partially  drowned  orifice.  Velocity  of  approach.  Coefficient  of  resistance. 
Sudden  enlargement  of  a  current  of  water.  Sudden  contraction  of  a 
current  of  water.  Loss  of  head  due  to  sharp-edged  entrance  into  a  pipe  or 
mouthpiece.  Mouthpieces.  Borda's  mouthpiece.  Conical  mouthpieces 
and  nozzles.  Flow  through  orifices  and  mouthpieces  under  constant 
pressure.  Time  of  emptying  a  tank  or  reservoir.  Notches  and  weirs. 
Rectangular  sharp-edged  weir.  Derivation  of  the  weir  formula  from  that 
of  a  large  orifice.  Thomson's  principle  of  similarity.  Discharge  through 
a  trianglar  notch  by  the  principle  of  similarity.  Discharge  through  a 
rectangular  weir  by  the  principle  of  similarity.  Rectangular  weir  with 
end  contractions.  Bazin's  formula  for  the  discharge  of  a  weir.  Bazin's 
and  the  Cornell  experiments  on  weirs.  Velocity  of  approach.  Influence  of 
the  height  of  the  weir  sill  above  the  bed  of  the  stream  on  the  contraction. 
Discharge  of  a  weir  when  the  air  is  not  freely  admitted  beneath  the  nappe. 
Form  of  the  nappe.  Depressed  nappe.  Adhering  nappes.  Drowned  or 
wetted  nappes.  Instability  of  the  form  of  the  nappe.  Drowned  weirs  with 
sharp  crests.  Vertical  weirs  of  small  thickness.  Depressed  and  wetted 
nappes  for  flat-crested  weirs.  Drowned  nappes  for  flat-crested  weirs.  Wide 
flat-crested  weirs.  Flow  over  dams.  Form  of  weir  for  accurate  gauging. 
Boussinesq's  theory  of  the  discharge  over  a  weir.  Determining  by  ap- 
proximation the  discharge  of  a  weir,  when  the  velocity  of  approach  is 
unknown.  Time  required  to  lower  the  water  in  a  reservoir  a  given  distance 
by  means  of  a  weir.  Examples  .......  Page  50 


CHAPTER  V. 

FLOW    THROUGH    PIPES. 

Resistances  to  the  motion  of  a  fluid  in  a  pipe.  Loss  of  head  by  friction. 
Head  lost  at  the  entrance  to  the  pipe.  Hydraulic  gradient  and  virtual 
slope.  Determination  of  the  loss  of  head  due  to  friction.  Reynold's 
apparatus.  Equation  of  flow  in  a  pipe  of  uniform  diameter  and  determi- 
nation of  the  head  lost  due  to  friction.  Hydraulic  mean  depth.  Empirical 


CONTENTS  ix 

formulae  for  loss  of  head  due  to  friction.  Formula  of  Darcy.  Variation 
of  C  in  the  formula  v  =  G\fmi  with  service.  Ganguillet  and  Kutter's 
formula.  Reynold's  experiments  and  the  logarithmic  formula.  Critical 
velocity.  Critical  velocity  by  the  method  of  colour  bands.  Law  of 
frictional  resistance  for  velocities  above  the  critical  velocity.  The  de- 
termination of  the  values  of  C  given  in  Table  XII.  Variation  of  fc,  in  the 
formula  i  =  kvn,  with  the  diameter.  Criticism  of  experiments.  Piezometer 
fittings.  Effect  of  temperature  on  the  velocity  of  flow.  Loss  of  head  due 
to  bends  and  elbows.  Variations  of  the  velocity  at  the  cross  section  of  a 
cylindrical  pipe.  Head  necessary  to  give  the  mean  velocity  vm  to  the 
water  in  the  pipe.  Practical  problems.  Velocity  of  flow  in  pipes.  Trans- 
mission of  power  along  pipes  by  hydraulic  pressure.  The  limiting  diameter 
of  cast  iron  pipes.  Pressures  on  pipe  bends.  Pressure  on  a  plate  in  a  pipe 
filled  with  flowing  water.  Pressure  on  a  cylinder.  Examples  .  Page  112 


CHAPTER  VI. 

FLOW   IN   OPEN   CHANNELS. 

Variety  of  the  forms  of  channels.  Steady  motion  in  uniform  channels* 
Formula  for  the  flow  when  the  motion  is  uniform  in  a  channel  of  uniform 
section  and  slope.  Formula  of  Chezy.  Formulae  of  Prony  and  Eytelwein. 
Formula  of  Darcy  and  Bazin.  Ganguillet  and  Kutter's  formula.  Bazin's 
formula.  Variations  of  the  coefficient  C.  Logarithmic  formula  for  flow  in 
channels.  Approximate  formula  for  the  flow  in  earth  channels.  Distribu- 
tion of  velocity  in  the  cross  section  of  open  channels.  Form  of  the  curve 
of  velocities  on  a  vertical  section.  The  slopes  of  channels  and  the  velocities 
allowed  in  them.  Sections  of  aqueducts  and  sewers.  Siphons  forming 
part  of  aqueducts.  The  best  form  of  channel.  Depth  of  flow  in  a  circular 
channel  for  maximum  velocity  and  maximum  discharge.  Curves  of  velocity 
and  discharge  for  a  channel.  Applications  of  the  formulae.  Problems. 
Examples Page  178 


CHAPTER   VII. 

GAUGING   THE   FLOW  OF  WATER. 

Measuring  the  flow  of  water  by  weighing.  Meters.  Measuring  the  flow 
by  means  of  an  orifice.  Measuring  the  flow  in  open  channels.  Surface 
floats.  Double  floats.  Rod  floats.  The  current  meter.  Pitot  tube.  Cali- 
bration of  Pitot  tubes.  Gauging  by  a  weir.  The  hook  gauge.  Gauging 
the  flow  in  pipes ;  Venturi  meter.  Deacon's  waste-water  meter.  Kennedy's 
meter.  Gauging  the  flow  of  streams  by  chemical  means.  Examples 

Page  234 


X  CONTENTS 

CHAPTER   VIII. 

IMPACT   OF   WATER   ON   VANES. 

Definition  of  vector.  Sum  of  two  vectors.  Resultant  of  two  velocities. 
Difference  of  two  vectors.  Impulse  of  water  on  vanes.  Relative  velocity. 
Definition  of  relative  velocity  as  a  vector.  To  find  the  pressure  on  a 
moving  vane,  and  the  rate  of  doing  work.  Impact  of  water  on  a  vane 
when  the  directions  of  motion  of  the  vane  and  jet  are  not  parallel. 
Conditions  which  the  vanes  of  hydraulic  machines  should  satisfy. 
Definition  of  angular  momentum.  Change  of  angular  momentum.  Two 
important  principles.  Work  done  on  a  series  of  vanes  fixed  to  a  wheel 
expressed  in  terms  of  the  velocities  of  whirl  of  the  water  entering  and 
leaving  the  wheel.  Curved  vanes.  Pelton  wheel.  Force  tending  to  move 
a  vessel  from  which  water  is  issuing  through  an  orifice.  The  propulsion 
of  ships  by  water  jets.  Examples Page  261 


CHAPTER   IX. 

WATER    WHEELS   AND   TURBINES. 

Overshot  water  wheels.  Breast  wheel.  Sagebien  wheels.  Impulse 
wheels.  Poncelet  wheel.  Turbines.  Reaction  turbines.  Outward  flow 
turbines.  Losses  of  head  due  to  frictional  and  other  resistances  in  outward 
flow  turbines.  Some  actual  outward  flow  turbines.  Inward  flow  turbines. 
Some  actual  inward  flow  turbines.  The  best  peripheral  velocity  for 
inward  and  outward  flow  turbines.  Experimental  determination  of  the 
best  peripheral  velocity  for  inward  and  outward  flow  turbines.  Value  of  e 

to  be  used  in  the  formula  —  =  eR.     The  ratio  of  the  velocity  of  whirl  V  to 

\J 

the  velocity  of  the  inlet  periphery  v.  The  velocity  with  which  water 
leaves  a  turbine.  Bernouilli's  equations  for  inward  and  outward  flow 
turbines  neglecting  friction.  Bernouilli's  equations  for  the  inward  and 
outward  flow  turbines  including  friction.  Turbine  to  develope  a  given 
horse-power.  Parallel  or  axial  flow  turbines.  Regulation  of  the  flow  to 
parallel  flow  turbines.  Bernouilli's  equations  for  axial  flow  turbines. 
Mixed  flow  turbines.  Cone  turbine.  Effect  of  changing  the  direction  of 
the  guide  blade,  when  altering  the  flow  of  inward  flow  and  mixed  flow 
turbines.  Effect  of  diminishing  the  flow  through  turbines  on  the  velocity 
of  exit.  Regulation  of  the  flow  by  means  of  cylindrical  gates.  The  Swain 
gate.  The  form  of  the  wheel  vanes  between  the  inlet  and  outlet  of 
turbines.  The  limiting  head  for  a  single  stage  reaction  turbine.  Series 
or  multiple  stage  reaction  turbines.  Impulse  turbines.  The  form  of  the 
vanes  for  impulse  turbines,  neglecting  friction.  Triangles  of  velocity  for 
an  axial  flow  impulse  turbine  considering  friction.  Impulse  turbine  for 
high  head.  Pelton  wheel.  Oil  pressure  governor  or  regulator.  Water 
pressure  regulators  for  impulse  turbines.  Hammer  blow  in  a  long  turbine 
supply  pipe.  Examples page  283 


CONTENTS  XI 

CHAPTER  X. 

PUMPS. 

Centrifugal  and  turbine  pumps.  Starting  centrifugal  or  turbine  pumps. 
Form  of  the  vanes  of  centrifugal  pumps.  Work  done  on  the  water  by  the 
wheel.  Ratio  of  velocity  of  whirl  to  peripheral  velocity.  The  kinetic  energy 
of  the  water  at  exit  from  the  wheel.  Gross  lift  of  a  centrifugal  pump. 
Efficiencies  of  a  centrifugal  pump.  Experimental  determination  of  the 
efficiency  of  a  centrifugal  pump.  Design  of  pump  to  give  a  discharge  Q. 
The  centrifugal  head  impressed  on  the  water  by  the  wheel.  Head-velocity 
curve  of  a  centrifugal  pump  at  zero  discharge.  Variation  of  the  discharge 
of  a  centrifugal  pump  with  the  head  when  the  speed  is  kept  constant. 
Bernouilli's  equations  applied  to  centrifugal  pumps.  Losses  in  centrifugal 
pumps.  Variation  of  the  head  with  discharge  and  with  the  speed  of  a 
centrifugal  pump.  The  effect  of  the  variation  of  the  centrifugal  head  and 
the  loss  by  friction  on  the  discharge  of  a  pump.  The  effect  of  the  diminu- 
tion of  the  centrifugal  head  and  the  increase  of  the  friction  head  as  the 
flow  increases,  on  the  velocity.  Discharge  curve  at  constant  head.  Special 

U2 
arrangements  for  converting  the  velocity  head  -^-  ,  with  which  the  water 

*9 

leaves  the  wheel,  into  pressure  head.  Turbine  pumps.  Losses  in  the 
spiral  casings  of  centrifugal  pumps.  General  equation  for  a  centrifugal 
pump.  The  limiting  height  to  which  a  single  wheel  centrifugal  pump  can 
be  used  to  raise  water.  The  suction  of  a  centrifugal  pump.  Series  or 
multi-stage  turbine  pumps.  Advantages  of  centrifugal  pumps.  Pump 
delivering  into  a  long  pipe  line.  Parallel  flow  turbine  pump.  Inward  flow 
turbine  pump.  Reciprocating  pumps.  Coefficient  of  discharge  of  the 
pump.  Slip.  Diagram  of  work  done  by  the  pump.  The  accelerations 
of  the  pump  plunger  and  the  water  in  the  suction  pipe.  The  effect  of 
acceleration  of  the  plunger  on  the  pressure  in  the  cylinder  during  the 
suction  stroke.  Accelerating  forces  in  the  delivery  pipe.  Variation  of 
pressure  in  the  cylinder  due  to  friction.  Air  vessel  on  the  suction  pipe. 
Air  vessel  on  the  delivery  pipe.  Separation  during  the  suction  stroke. 
Negative  slip.  Separation  in  the  delivery  pipe.  Diagram  of  work  done 
considering  the  variable  quantity  of  water  in  the  cylinder.  Head  lost  at 
the  suction  valve.  Variation  of  the  pressure  in  hydraulic  motors  due  to 
inertia  forces.  Worked  examples.  High  pressure  plunger  pump.  Tangye 
Duplex  pump.  The  hydraulic  ram.  Lifting  water  by  compressed  air. 
Examples Page  392 

CHAPTER  XL 

HYDRAULIC   MACHINES. 

Joints  and  packings  used  in  hydraulic  work.  The  accumulator.  Dif- 
ferential accumulator.  Air  accumulator.  Intensifies.  Steam  intensifiers. 
Hydraulic  forging  press.  Hydraulic  cranes.  Double  power  cranes. 
Hydraulic  crane  valves.  Hydraulic  press.  Hydraulic  riveter.  Brother- 
hood and  Rigg  hydraulic  engines.  Examples  ....  Page  485 


Xll  CONTENTS 

CHAPTER   XII. 

RESISTANCE   TO   THE   MOTION   OF   BODIES   IN   WATER. 

Froude's  experiments  on  the  resistance  of  thin  boards.  Stream  line 
theory  of  the  resistance  offered  to  motion  of  bodies  in  water.  Determination 
of  the  resistance  of  a  ship  from  that  of  the  model.  Examples  .  Page  507 

CHAPTER   XIII. 

STREAM   LINE  MOTION. 

Hele  Shaw's  experiments.  Curved  stream  line  motion.  Scouring  of 
river  banks  at  bends Page  517 

ANSWERS  TO  EXAMPLES    .        .        .        .        .        .        .        .    Page  521 

INDEX Page  525 


UNIVERSITY 

OF 


HYDRAULICS. 


CHAPTER  I. 

FLUIDS  AT   BEST. 


ERRATA. 

Page  25,  formula  3.     Insert  6  after  «•. 

„     82,  for  (L-O'lN)  substitute  (L-O'INH). 
Pages  122  and  148,  for  192  substitute  197. 
Page  309,  line  21,  substitute  cos£  for  cot/3. 

„      444,     „     29,          „          OC      „     OD. 

»>        »       »      »»  »>          OK      „    CK. 

„      4L  lines  19  and  20,  interchange  "  full  "  and  "  dotted." 


orifices  in  the  sides  of  tanks  and  through  short  pipes,  probably 
*  The  Aqueducts  of  Rome.     Frootinus,  translated  by  Herschel. 

L-B-  l 


Xli  CONTENTS 

CHAPTER   XII. 

RESISTANCE   TO   THE  MOTION   OF   BODIES   IN   WATER. 

Fronde's  experiments  on  the  resistance  of  thin  boards.  Stream  line 
theory  of  the  resistance  offered  to  motion  of  bodies  in  water.  Determination 
of  the  resistance  of  a  ship  from  that  of  the  model.  Examples  .  Page  507 

CHAPTER   XIII. 

STREAM   LINE   MOTION. 


XV^^JN, 

/  OF  THE  A 

f    UNIVERSITY  } 

OF  / 

\^IFC*^/ 


HYDRAULICS. 


CHAPTER  I. 

FLUIDS   AT   BEST. 

1.     Introduction. 

The  science  of  Hydraulics  in  its  limited  sense  as  originally 
understood,  had  for  its  object  the  consideration  of  the  laws 
regulating  the  flow  of  water  in  channels,  but  it  has  come  to 
have  a  wider  significance,  and  it  now  embraces,  in  addition,  the 
study  of  the  principles  involved  in  the  pumping  of  water  and  other 
fluids  and  their  application  to  the  working  of  different  kinds  of 
machines. 

The  practice  of  conveying  water  along  artificially  constructed 
channels  for  irrigation  and  domestic  purposes  dates  back  into 
great  antiquity.  The  Egyptians  constructed  transit  canals  for 
warlike  purposes,  as  early  as  3000  B.C.,  and  works  for  the  better 
utilisation  of  the  waters  of  the  Nile  were  carried  out  at  an  even 
earlier  date.  According  to  Josephus,  the  gardens  of  Solomon 
were  made  beautiful  by  fountains  and  other  water  works.  The 
aqueducts  of  Rome*,  some  of  which  were  constructed  more  than 
2000  years  ago,  were  among  the  "  wonders  of  the  world,"  and 
to-day  the  city  of  Athens  is  partially  supplied  with  water  by 
means  of  an  aqueduct  constructed  probably  some  centuries  before 
the  Christian  era. 

The  science  of  Hydraulics,  however,  may  be  said  to  have  only 
come  into  existence  at  the  end  of  the  seventeenth  century  when 
the  attention  of  philosophers  was  drawn  to  the  problems  involved 
in  the  design  of  the  fountains,  which  came  into  considerable  use 
in  Italian  landscape  gardens,  and  which,  according  to  Bacon, 
were  of  "great  beauty  and  refreshment."  The  founders  were 
principally  Torricelli  and  Marriott  from  the  experimental,  and 
Bernouilli  from  the  theoretical,  side.  The  experiments  of  Torri- 
celli and  of  Marriott  to  determine  the  discharge  of  water  through 
orifices  in  the  sides  of  tanks  and  through  short  pipes,  probably 

*  The  Aqueducts  of  Rome.     Frontinus,  translated  by  Herschel. 
L.  H.  1 


2  HYDRAULICS 

mark  the  first  attempts  to  determine  the  laws  regulating  the 
flow  of  water,  and  Torricelli's  famous  theorem  may  be  said  to 
be  the  foundation  of  modern  Hydraulics.  But,  as  shown  in  the 
chapter  on  the  flow  of  water  in  pipes,  it  was  not  until  a  century 
later  that  any  serious  attempt  was  made  to  give  expression  to  the 
laws  regulating  the  flow  in  long  pipes  and  channels,  and  practi- 
cally the  whole  of  the  knowledge  we  now  possess  has  been 
acquired  during  the  last  century.  Simple  machines  for  the 
utilisation  of  the  power  of  natural  streams  have  been  made  for 
many  centuries,  examples  of  which  are  to  be  found  in  an  interest- 
ing work  Hydrostatics  and  Hydraulics  written  in  English  by 
Stephen  Swetzer  in  1729,  but  it  has  been  reserved  to  the  workers 
of  the  nineteenth  century  to  develope  all  kinds  of  hydraulic 
machinery,  and  to  discover  the  principles  involved  in  their  correct 
design.  Poncelet's  enunciation  of  the  correct  principles  which 
should  regulate  the  design  of  the  "floats"  or  buckets  of  water 
wheels,  and  Fourneyron's  application  of  the  triangle  of  velocities 
to  the  design  of  turbines,  marked  a  distinct  advance,-  but  it  must 
be  admitted  that  the  enormous  development  of  this  class  of 
machinery,  and  the  very  high  standard  of  efficiency  obtained,  is 
the  outcome,  not  of  theoretical  deductions,  but  of  experience, 
and  the  careful,  scientific  interpretation  of  the  results  of 
experiments. 

2.     Fluids  and  their  properties. 

The  name  fluid  is  given,  in  general,  to  a  body  which  offers 
very  small  resistance  to  deformation,  and  which  takes  the  shape 
of  the  body  with  which  it  is  in  contact. 

If  a  solid  body  rests  upon  a  horizontal  plane,  a  force  is  required 
to  move  the  body  over  the  plane,  or  to  overcome  the  friction 
between  the  body  and  the  plane.  If  the  plane  is  very  smooth 
the  force  may  be  very  small,  and  if  we  conceive  the  plane  to  be 
perfectly  smooth  the  smallest  imaginable  force  would  move  the 
body. 

If  in  a  fluid,  a  horizontal  plane  be  imagined  separating  the 
fluid  into  two  parts,  the  force  necessary  to  cause  the  upper 
part  to  slide  over  the  lower  will  be  very  small  indeed,  and 
any  force,  however  small,  applied  to  the  fluid  above  the  plane 
and  parallel  to  it,  will  cause  motion,  or  in  other  words  will  cause 
a  deformation  of  the  fluid. 

Similarly,  if  a  very  thin  plate  be  immersed  in  the  fluid  in  any 
direction,  the  plate  can  be  made  to  separate  the  fluid  into  two 
parts  by  the  application  to  the  plate  of  an  infinitesimal  force, 
and  in  the  imaginary  perfect  fluid  this  force  would  be  zero. 


FLUIDS  AT  REST  3 

Viscosity.  Fluids  found  in  nature  are  not  perfect  and  are 
said  to  have  viscosity;  but  when  they  are  at  rest  the  conditions 
of  equilibrium  can  be  obtained,  with  sufficient  accuracy,  on 
the  assumption  that  they  are  perfect  fluids,  and  that  therefore 
no  tangential  stresses  can  exist  along  any  plane  in  a  fluid. 
This  branch  of  the  study  of  fluids  is  called  Hydrostatics ;  when 
the  laws  of  movement  of  fluids  are  considered,  as  in  Hydraulics, 
these  tangential,  or  frictional  forces  have  to  be  taken  into 
consideration. 

3.  Compressible  and  incompressible  fluids. 

There  are  two  kinds  of  fluids,  gases  and  liquids,  or  those  which 
are  easily  compressed,  and  those  which  are  compressed  with 
difficulty.  The  amount  by  which  the  volumes  of  the  latter  are 
altered  for  a  very  large  variation  in  the  pressure  is  so  small  that 
in  practical  problems  this  variation  is  entirely  neglected,  and 
they  are  therefore  considered  as  incompressible  fluids. 

In  this  volume  only  incompressible  fluids  are  considered,  and 
attention  is  confined,  almost  entirely,  to  the  one  fluid,  water. 

4.  Density  and  specific  gravity. 

The  density  of  any  substance  is  the  weight  of  unit  volume  at 
the  standard  temperature  and  pressure. 

The  specific  gravity  of  any  substance  at  any  temperature  and 
pressure  is  the  ratio  of  the  weight  of  unit  volume  to  the  weight 
of  unit  volume  of  pure  water  at  the  standard  temperature  and 
pressure. 

The  variation  of  the  volume  of  liquid  fluids,  with  the  pressure, 
as  stated  above,  is  negligible,  and  the  variation  due  to  changes  of 
temperature,  such  as  are  ordinarily  met  with,  is  so  small,  that  in 
practical  problems  it  is  unnecessary  to  take  it  into  account. 

In  the  case  of  water,  the  presence  of  salts  in  solution  is  of 
greater  importance  in  determining  the  density  than  variations 
of  temperature,  as  will  be  seen  by  comparing  the  densities  of  sea 
water  and  pure  water  given  in  the  following  table. 

TABLE  I. 

Useful  data. 

One  cubic  foot  of  water  at  39-1°  F.  weighs  62-425  Ibs. 
60°  F.          „       62-36     „ 

One  cubic  foot  of  average  sea  water  at  60°  F.  weighs  64  Ibs. 
One  gallon  of  pure  water  at  60°  F.  weighs  10  Ibs. 
One  gallon  of  pure  water  has  a  volume  of  277'25  cubic  inches. 
One  ton  of  pure  water  at  60°  F.  has  a  volume  of  35-9  cubic  feet. 

1—2 


4  HYDRAULICS 

Table  of  densities  of  pure  water. 

Temperature 

Degrees  Fahrenheit  Density 
32  -99987 

39-1  1-000000 

50  0-99973 

60  0-99905 

80  0-99664 

104  0-99233 

From  the  above  it  will  be  seen  that  in  practical  problems  it 
will  be  sufficiently  near  to  take  the  weight  of  one  cubic  foot  of 
fresh  water  as  62*4  Ibs.,  one  gallon  as  10  pounds,  6'24  gallons  in  a 
cubic  foot,  and  one  cubic  foot  of  sea  water  as  64  pounds. 

5.  Hydrostatics. 

A  knowledge  of  the  principles  of  hydrostatics  is  very  helpful 
in  approaching  the  subject  of  hydraulics,  and  in  the  wider  sense 
in  which  the  latter  word  is  now  used  it  may  be  said  to  include  the 
former.  It  is,  therefore,  advisable  to  consider  the  laws  of  fluids 
at  rest. 

There  are  two  cases  to  consider.  First,  fluids  at  rest  under  the 
action  of  gravity,  and  second,  those  cases  in  which  the  fluids  are 
at  rest,  or  are  moving  very  slowly,  and  are  contained  in  closed 
vessels  in  which  pressures  of  any  magnitude  act  upon  the  fluid, 
as,  for  instance,  in  hydraulic  lifts  and  presses. 

6.  Intensity  of  pressure. 

The  intensity  of  pressure  at  any  point  in  a  fluid  is  the  pressure 
exerted  upon  unit  area,  if  the  pressure  on  the  unit  area  is  uniform 
and  is  exerted  at  the  same  rate  as  at  the  point. 

Consider  any  little  element  of  area  a,  about  a  point  in  the  fluid,, 
and  upon  which  the  pressure  is  uniform. 

If  P  is  the  total  pressure  on  a,  the  Intensity  of  Pressure  p,  is  then 

P 

P=a> 
or  when  P  and  a  are  indefinitely  diminished, 

3P 
P="da' 

7.  The  pressure  at  any  point  in  a  fluid  is  the  same  in  all 
directions. 

It  has  been  stated  above  that  when  a  fluid  is  at  rest  its  resist- 
ance to  lateral  deformation  is  practically  zero  and  that  on  any 
plane  in  the  fluid  tangential  stresses  cannot  exist.  From  this 
experimental  fact  it  follows  that  the  pressure  at  any  point  in  the 
fluid  is  the  same  in  all  directions. 


FLUIDS  AT  REST  5 

Consider  a  small  wedge  ABC,  Fig.  1,  floating  immersed  in  a 
fluid  at  rest. 

Since   there   cannot  be   a  tangential  i^ 

stress  on  any  of  the  planes  AB,  BC,  or  AC,     Av.    A ^- .B 

the  pressures  on  them  must  be  normal. 


Let  p,  P!  and  p2  be  the  intensities  of  -p^ 

pressures  on  these  planes  respectively. 

The  weight  of  the  wedge  will  be  very  Fig.  1. 

small  and  may  be  neglected. 

As  the  wedge  is  in  equilibrium  under  the  forces  acting  on 
its  three  faces,  the  resolved  components  of  the  force  acting  on 
AC  in  the  directions  of  p  and  pl  must  balance  the  forces  acting 
on  AB  and  BC  respectively. 

Therefore  p.2 .  AC  cos  0  =  p .  AB, 

and  puACsinfl^^BC. 

But  AB  =  ACcos0, 

and  BC  =  ACsin0. 

Therefore  p  =  pl  =  p^ . 

8.  The  pressure  on  any  horizontal  plane  in  a  fluid  must 
be  constant. 

Consider  a  small  cylinder  of  a  fluid  joining  any  two  points  A 
and  B  on  the  same  horizontal  plane  in  the  fluid. 

Since  there  can  be  no  tangential  forces  acting  on  the  cylinder 
parallel  to  the  axis,  the  cylinder  must  be  in  equilibrium  under  the 
pressures  on  the  ends  A  and  B  of  the  cylinder,  and  since  these 
are  of  equal  area,  the  pressure  must  be  the  same  at  each  end  of 
the  cylinder. 

9.  Fluids  at  rest,  with  the  free  surface  horizontal. 

The  pressure  per  unit  area  at  any  depth  h  below  the  free 
surface  of  a  fluid  due  to  the  weight  of  the  fluid  is  equal  to  the 
weight  of  a  column  of  fluid  of  height  h  and  of  unit  sectional  area. 

Let  the  pressure  per  unit  area  acting  on  the  surface  of  the 
fluid  be  p  Ibs.  If  the  fluid  is  in  a  closed  vessel,  the  pressure  p  may 
have  any  assigned  value,  but  if  the  free  surface  is  exposed  to  the 
atmosphere,  p  will  be  the  atmospheric  pressure. 

If  a  small  open  tube  AB,  of  length  h,  and  cross  sectional  area  a, 
be  placed  in  the  fluid,  the  weight  per  unit  volume  of  which  is 
w  Ibs.,  with  its  axis  vertical,  and  its  upper  end  A  coincident  with 
the  surface  of  the  fluid,  the  weight  of  fluid  in  the  cylinder  must  be 
w .  a .  h  Ibs.  The  pressure  acting  on  the  end  A  of  the  column 
is  pa  Ibs. 


6 


HYDRAULICS 


Since  there  cannot  be  any  force  acting  on  the  column  parallel 
to  the  sides  of  the  tube,  the  force  of  wah  Ibs.  +  pa  Ibs.  must  be 
kept  in  equilibrium  by  the  pressure  of  the  external  fluid  acting  on 
the  fluid  in  the  cylinder  at  the  end  B. 

The  pressure  per  unit  area  at  B,  therefore, 

wah  +  pa     ,    , 
= • —  =  (wfi  +  p)  Ibs. 

The  pressure  per  unit  area,  therefore,  due  to  the  weight  of  the 
fluid  only  is  wh  Ibs. 

In  the  case  of  water,  w  may  be  taken  as  62*40  Ibs.  per  cubic 
foot  and  the  pressure  per  sq.  foot  at  a  depth  of  h  feet  is,  therefore, 
62'407i  Ibs.,  and  per  sq.  inch  '433/1  Ibs. 

It  should  be  noted  that  the  pressure  is  independent  of  the  form 
of  the  vessel,  and  simply  depends  upon  the  vertical  depth  of  the 
point  considered  below  the  surface  of  the  fluid.  This  can  be 
illustrated  by  the  different  vessels  shown  in  Fig.  2.  If  these 
were  all  connected  together  by  means  of  a  pipe,  the  fluid  when 
at  rest  would  stand  at  the  same  level  in  all  of  them,  and  on  any 
horizontal  plane  AB  the  pressure  would  be  the  same. 


D 


B 


Pressure  an,  the  Plaite  A  B= w-h  Ibs  per  sq  Foot. 
Fig.  2. 

If  now  the  various  vessels  were  sealed  from  each  other 
by  closing  suitable  valves,  and  the  pipe  taken  away  without 
disturbing  the  level  CD  in  any  case,  the  intensity  of  pressure  on 
AB  would  remain  unaltered,  and  would  be,  in  all  cases,  equal 
to  wh. 

Example.  In  a  condenser  containing  air  and  water,  the  pressure  of  the  air  is 
2  Ibs.  per  sq.  inch  absolute.  Find  the  pressure  per  sq.  foot  at  a  point  3  feet  below 
the  free  surface  of  the  water. 

^  =  2x144  +  3x62-4 
=  475-2  Ibs.  per  sq.  foot. 


FLUIDS   AT   REST  7 

10.  Pressures  measured  in  feet  of  water.    Pressure  head. 
It  is  convenient  in  hydrostatics  and  hydraulics  to  express  the 

pressure  at  any  point  in  a  fluid  in  feet  of  the  fluid  instead  of  pounds 
per  sq.  foot  or  sq.  inch.  It  follows  from  the  previous  section  that 
if  the  pressure  per  sq.  foot  is  p  Ibs.  the  equivalent  pressure  in  feet 

of  water,  or  the  pressure  head,  is  h  =  ^  ft.  and  for  any  other  fluid 
having  a  specific  gravity  p,  the  pressure  per  sq.  foot  for  a  head 

h  of  the  fluid  is  -»  =  w  .  p  .  h,  or  h  =  —  . 

wp 

11.  Piezometer  tubes. 

The  pressure  in  a  pipe  or  other  vessel  can  conveniently  be 
measured  by  fixing  a  tube  in  the  pipe  and  noting  the  height  to 
which  the  water  rises  in  the  tube. 

Such  a  tube  is  called  a  pressure,  or  piezometer,  tube. 

The  tube  need  not  be  made  straight  but  may  be  bent  into  any 
form  and  carried,  within  reasonable  limits,  any  distance  horizon- 
tally. 

The  vertical  rise  h  of  the  water  will  be  always 

h=p, 

w' 
where  p  is  the  pressure  per  sq.  foot  in  the  pipe. 

If  instead  of  water,  a  liquid  of  specific  gravity  p  is  used  the 
height  h  to  which  the  liquid  will  rise  in  the  tube  is 


w  .p 

Example.  A  tube  having  one  end  open  to  the  atmosphere  is  fitted  into  a  pipe 
containing  water  at  a  pressure  of  10  Ibs.  per  sq.  inch  above  the  atmosphere.  Find 
the  height  to  which  the  water  will  rise  in  the  tube. 

The  water  will  rise  to  such  a  height  that  the  pressure  at  the  end  of  the  tube  in 
the  pipe  due  to  the  column  of  water  will  be  10  Ibs.  per  sq.  inch. 

Therefore  h=  10  X  144  =  23  -08  feet. 

w 

12.     The  barometer. 

The  method  of  determining  the  atmospheric 
pressure  by  means  of  the  barometer  can  now  be 
understood. 

If  a  tube  about  3  feet  long  closed  at  one  end  be 
completely  filled  with  mercury,  Fig.  3,  and  then 
turned  into  a  vertical  position  with  its  open  end 
in  a  vessel  containing  mercury,  the  liquid  in  the 
tube  falls  until  the  length  h  of  the  column  is  about 
30  inches  above  the  surface  of  the  mercury  in  the 
vessel. 


HYDRAULICS 


Since  the  pressure  p  on  the  top  of  the  mercury  is  now  zero,  the 
pressure  per  unit  area  acting  on  the  section  of  the  tube,  level  with 
the  surface  of  the  mercury  in  the  vessel,  must  be  equal  to  the 
weight  of  a  column  of  mercury  of  height  h. 

The  specific  gravity  of  the  mercury  is  13'596  at  the  standard 
temperature  and  pressure,  and  therefore  the  atmospheric  pressure 
per  sq.  inch,  pa,  is, 

30"  x  13'596  x  62'4 
Pa  =  -        10..  tAA        -  -  14  7  Ibs.  per  sq.  inch. 


12  x  144 
Expressed  in  feet  of  water, 

_  14-7  x  144 
62-4 


-  33-92  feet. 


This  is  so  near  to  34  feet  that  for  the  standard  atmospheric 
pressure  this  value  will  be  taken  throughout  this  book. 

A  similar  tube  can  be  conveniently  used  for  measuring  low 
pressures,  lighter  liquids  being  used  when  a  more  sensitive  gauge 
is  required. 

13.     The  differential  gauge. 

A  more  convenient  arrangement  for  measuring  pressures,  and 
one  of  considerable  utility  in  many  hydraulic  experiments,  is 
known  as  the  differential  gauge. 

Let  ABCD,  Fig.  4,  be  a  simple  U  tube 
containing  in  the  lower  part  some  fluid  of 
known  density. 

If  the  two  limbs  of  the  tube  are  open  to 
the  atmosphere  the  two  surfaces  of  the  fluid 
will  be  in  the  same  horizontal  plane. 

If,  however,  into  the  limbs  of  the  tube  a 
lighter  fluid,  which  does  not  mix  with  the 
lower  fluid,  be  poured  until  it  rises  to  C  in 
one  tube  and  to  D  in  the  other,  the  two 
surfaces  of  the  lower  fluid  will.,  now  be  at 
different  levels. 

Let  B  and  E  be  the  common  surfaces  of 
the   two   fluids,  h  being   their   difference  of 
level,  and  hi  and  h2  the  heights  of  the  free 
surfaces  of  the  lighter  fluid  above  E  and  B  respectively. 

Let  p  be  the  pressure  of  the  atmosphere  per  unit  area,  and  d 
and  di  the  densities  of  the  lower  and  upper  fluids  respectively. 
Then,  since  upon  the  horizontal  plane  AB  the  fluid  pressure  must 
be  constant, 

p  +  dih2  =  p  +  dihi  +  dh, 
or  di  (h2  -  hi)  =  dh. 


Q 

^ 
c 

V 

D 

T~ 

Cs» 

< 

r*J 

F 

f 

L         A 

ll 

Fig.  4. 


FLUIDS  AT  REST 


9 


If  now,  instead  of  the  two  limbs  of  the  U  tube  being  open  to 
the  atmosphere,  they  are  connected  by  tubes  to  closed  vessels  in 
which  the  pressures  are  pl  and  p.2  pounds  per  sq.  'foot  -respectively, 
and  /ij  and  /^  are  the  vertical  lengths  of  the  columns  of  fluid  above 
E  and  B  respectively,  then 


.  h, 


or 


An  application  of  such  a  tube  to  determine  the  difference  of 
pressure  at  two  points  in  a  pipe  containing  flowing  water  is  shown 
in  Fig.  88,  page  116. 

Fluids  generally  used  in  such  U  tubes.  In  hydraulic  experiments 
the  upper  part  of  the  tube  is  filled  with  water,  and  therefore  the 
fluid  in  the  lower  part  must  have  a  greater  density  than  water. 
When  the  difference  of  pressure  is  fairly  large,  mercury  is  generally 
used,  the  specific  gravity  of  which  is  13*596.  When  the  difference 
of  pressure  is  small,  the  height  h  is  difficult  to  measure  with 
precision,  so  that,  if  this  form  of  gauge  is  to  be  used,  it  is  desirable 
to  replace  the  mercury  by  a  lighter  liquid.  Carbon  bisulphide 
has  been  used  but  its  action  is  sluggish  and  the  meniscus  between 
it  and  the  water  is  not  always  well  defined. 
Nitro-benzine  gives  good  results,  its  prin- 
cipal fault  being  that  the  falling  meniscus 
does  not  very  quickly  assume  a  definite 
shape. 

The  inverted  air  gauge.  A  more  sen- 
sitive gauge  can  be  made  by  inverting  a 
U  tube  and  enclosing  in  the  upper  part 
a  certain  quantity  of  air  as  in  the  tube 
BHC,  Fig.  5. 

Let  the  pressure  at  D  in  the  limb  DF 
be  pi  pounds  peM  square  foot,  equivalent 
to  a  head  hi  of  the  fluid  in  the  lower  part 
of  the  gauge,  and  at  A  in  the  limb  AE  let 
the  pressure  be  p2,  equivalent  to  a  head  7i2. 
Let  h  be  the  difference  of  level  of  G  and  C. 

Then  if  CGH  contains  air,  and  the  weight  of  the  air  be 
neglected,  being  very  small,  the  pressure  at  C  must  equal  the 
pressure  at  G  ;  and  since  in  a  fluid  the  pressure  on  any  horizontal 
plane  is  constant  the  pressure  at  C  is  equal  to  the  pressure  at  D, 
and  the  pressure  at  A  equal  to  the  pressure  at  B.  Again  the 
pressure  at  G-  is  equal  to  the  pressure  at  K. 

Therefore  h2-h  =  hi, 

•  h. 


g= 

H 

^ 

F 

fr 

~f 

-C 

^. 

E 

i 

G 

1  —  T" 

f 

F 

A 

L 

Bf  C 

L 

D 

Fig.  5. 


or 


-  Pi  = 


10 


HYDRAULICS 


If  the  fluid  is  water  p  may  then  be  taken  as  unity ;  for  a  given 
difference  of  pressure  the  value  of  h  will  clearly  be  much  greater 
than  for  the  mercury  gauge,  and  it  has  the  further  advantage  that 
In  gives  directly  the  difference  of  pressure  in  feet  of  water.  The 
temperature  of  the  air  in  the  tube  does  not  affect  the  readings,  as 
any  rise  in  temperature  will  simply  depress  the  two  columns 
without  affecting  the  value  of  h. 

The  inverted  oil  gauge.     A   still  more   sensitive   gauge    can 
however  be  obtained  by  using,  in  the 
upper  part  of  the  tube,  an  oil  lighter 
than  water  instead  of  air,  as  shown 
in  Fig.  6. 

Let  pi  and  p2  be  the  pressures  in 
the  two  limbs  of  the  tube  on  a  given 
horizontal  plane  AB,  hi  and  h2  being 
the  equivalent  heads  of  water.  The 
oil  in  the  bent  tube  will  then  take  up 
some  such  position  as  shown,  the 
plane  AD  being  supposed  to  coincide 
with  the  lower  surface  C. 

Then,  since  upon  any  horizontal 
plane  in  a  homogeneous  fluid  the 
pressure  must  be  constant,  the  pres- 
sures at  Gr  and  H  are  equal  and  also 
those  at  D  and  C. 

Let  PI  be  the  specific  gravity  of 
the  water,  and  p  of  the  oil. 

Then  Pihi-ph  =  pi  (fe  —  h). 

Therefore     h  (PI  -  p)  =  pi  (h2  -  hi) 


/ 

E 

\ 

~T" 

if- 

j 

G 

H 

,* 

k 

B    |C 

D 

\  ^ 

v^ 

Fig.  6. 


and 


(p.- 


(1) 


Substituting  for  hi  and  h2  the  values 


and 


(2), 


to .  Oh  -  P) 

or  p2  —  PJ  =  <£#  .  (pj  —  p)  /j,  (3) . 

From  (2)  it  is  evident  that,  if  the  density  of  the  oil  is  not  very 
different  from  that  of  the  water,  h  may  be  large  for  very  small 
differences  of  pressure.     Williams,  Hubbell  and  Fenkell*  found 
*  Proceedings  Am.S.C.E.,  Vol.  xxvii.  p.  384. 


FLUIDS   AT  REST 


11 


that  either  kerosene,  gasoline,  or  sperm  oil  gave  excellent  results, 
but  sperm  oil  was  too  sluggish  in  its  action  for  rapid  work. 
Kerosene  gave  the  best  results. 

Temperature  coefficient  of  the  inverted  oil  gauge.  Unlike  the 
inverted  air  gauge  the  oil  gauge  has  a  considerable  temperature 
coefficient,  as  will  be  seen  from  the  table  of  specific  gravities  at 
various  temperatures  of  water  and  the  kerosene  and  gasoline  used 
by  Williams,  Hubbell  and  Fenkell. 

In  this  table  the  specific  gravity  of  water  is  taken  as  unity 
at  60°  F. 


Temperature  °F. 
Specific  gravity 


Water 

Kerosene 

Gasoline 

40 
1-00092 

60        100 
1-0000  '9941 

40        60       100 
•7955  -7879  '7725 

40         60         80 
•72147  -71587  '70547 

The  calibration  of  the  inverted  oil  gauge.  Messrs  Williams, 
Hubbell  and  Fenkell  have  adopted  an  ingenious  method  of 
calibrating  the  oil  gauge.  This  will  readily  be  understood  on 
reference  to  Fig.  6. 

The  difference  of  level  of  E  and  F  clearly  gives  the  difference 
of  head  acting  on  the  plane  AD  in  feet  of  water,  and  this  from 

equation  (1)  equals  —        — . 

Water  is  put  into  AE  and  FD  so  that  the  surfaces  E  and  F 
are  on  the  same  level,  the  common  surfaces  of  the  oil  and  the 
water  also  being  on  the  same  level,  this  level  being  zero  for  the 
oil.  Water  is  then  run  out  of  FD  until  the  surface  F  is 
exactly  1  inch  below  E  and  a  reading  for  h  taken.  The  surface  F 
is  again  lowered  1  inch  and  a  reading  of  h  taken.  This  process 
is  continued  until  F  is  lowered  as  far  as  convenient,  and  then 
the  water  in  EA  is  drawn  out  in  a  similar  manner.  When  E 
and  F  are  again  level  the  oil  in  the  gauge  should  read  zero. 

14.    Transmission  of  fluid  pressure. 

If  an  external  pressure  be  applied  at  any  point  in  a  fluid,  it  is 
transmitted  equally  in  all  direc- 
tions through  the  whole  mass. 
This  is  proved  experimentally 
by  means  of  a  simple  apparatus 
such  as  shown  in  Fig.  7. 

If  a  pressure  P  is  exerted  upon 
a  small  piston  Q  of  a  sq.  inches 


« 


Fig.  7. 


12  HYDRAULICS 

p 

area,  the  pressure  per  unit  area  p  =  —  ,  and  the  piston  at  R  on  the 

a 

same  level  as  Q,  which  has  an  area  A,  can  be  made  to  lift  a  load  W 

p 
equal  to  A  —  ;  or  the  pressure  per  sq.  inch  at  R  is  equal  to  the 

CL 

pressure  at  Q.  The  piston  at  R  is  assumed  to  be  on  the  same  level 
as  Q  so  as  to  eliminate  the  consideration  of  the  small  differences  of 
pressure  due  to  the  weight  of  the  fluid. 

If  a  pressure  gauge  is  fitted  on  the  connecting  pipe  at  any 
point,  and  p  is  so  large  that  the  pressure  due  to  the  weight  of  the 
fluid  may  be  neglected,  it  will  be  found  that  the  intensity  of 
pressure  is  p.  This  result  could  have  been  anticipated  from  that 
of  section  8. 

Upon  this  simple  principle  depends  the  fact  that  enormous 
forces  can  be  exerted  by  means  of  hydraulic  pressure. 

If  the  piston  at  Q  is  of  small  area,  while  that  at  R  is  large, 
then,  since  the  pressure  per  sq.  inch  is  constant  throughout  the 
fluid, 

W     A 
P      a' 

or  a  very  large  force  W  can  be  overcome  by  the  application  of 
a  small  force  P.  A  very  large  mechanical  advantage  is  thus 
obtained. 

It  should  be  clearly  understood  that  the  rate  of  doing  work 
at  W,  neglecting  any  losses,  is  equal  to  that  at  P,  the  distance 
moved  through  by  W  being  to  that  moved  through  by  P  in 
the  ratio  of  P  to  W,  or  in  the  ratio  of  a  to  A. 

Example.  A  pump  ram  has  a  stroke  of  3  inches  and  a  diameter  of  1  inch.  The 
pump  supplies  water  to  a  lift  which  has  a  ram  of  5  inches  diameter.  The  force 
driving  the  pump  ram  is  1500  Ibs.  Neglecting  all  losses  due  to  friction  etc., 
determine  the  weight  lifted,  the  work  done  in  raising  it  5  feet,  and  the  number 
of  strokes  made  by  the  pump  while  raising  the  weight. 

Area  of  the  pump  ram  =  -7854  sq.  inch. 

Area  of  the  lift  ram=  19'6  sq.  inches. 


Therefore  W  =  ,=  37,500  Ibs. 

•78o4 

Work  done  =  37,500  x  5  =  187,500  ft.  Ibs. 

Let  N  equal  the  number  of  strokes  of  the  pump  ram. 
Then  N  x  T^  x  1500  Ibs.  =  187,500  ft.  Ibs. 

and  N  =  500  strokes. 

15.    Total  or  whole  pressure. 

The  whole  pressure  acting  on  a  surface  is  the  sum  of  all  the 
normal  pressures  acting  on  the  surface.  If  the  surface  is  plane  all 
the  forces  are  parallel,  and  the  whole  pressure  is  the  sum  of  these 
parallel  forces. 


FLUIDS    AT   REST 


13 


Let  any  surface,  which  need  not  be  a  plane,  be  immersed 
in  a  fluid.  Let  A  be  the  area  of  the  wetted  surface,  and  h  the 
pressure  head  at  the  centre  of  gravity  of  the  area.  If  the  area 
is  immersed  in  a  fluid  the  pressure  on  the  surface  of  which  is  zero, 
the  free  surface  of  the  fluid  will  be  at  a  height  h  above  the  centre 
of  gravity  of  the  area.  In  the  case  of  the  area  being  immersed  in 
a  fluid,  the  surface  of  which  is  exposed  to  a  pressure  p,  and  below 
which  the  depth  of  the  centre  of  gravity  of  the  area  is  7t0,  then 


w 


If  the  area  exposed  to  the  fluid  pressure  is  one  face  of  a  body, 
the  opposite  face  of  which  is  exposed  to  the  atmospheric  pressure, 
as  in  the  case  of  the  side  of  a  tank  containing  water,  or  the 
masonry  dam  of  Fig.  14,  or  a  valve  closing  the  end  of  a  pipe  as 
in  Fig.  8,  the  pressure  due  to  the 
atmosphere  is  the  same  on  the  two 
faces  and  therefore  may  be  neglected. 

Let  w  be  the  weight  of  a  cubic 
foot  of  the  fluid.  Then,  the  whole 
pressure  on  the  area  is 


If  the  surface  is  in  a  horizontal 
plane  the  theorem  is  obviously  true, 
since  the  intensity  of  pressure  is  con- 
stant and  equals  w  .  h. 

In   general,   imagine   the    surface, 


B 


Fig.  8. 


Fig.  9,  divided  into  a  large  number  of  small  areas  a,  Oi,  a^  ... . 

Let  x  be  the  depth  below  the  free  surface  FS,  of  any  element 
of  area  a ;  the  pressure  on  this  element  =  w .  x .  a. 

The  whole  pressure  P  =  ^w .x.a. 

But  w  is  constant,  and  the  sum  of  the  moments  of  the  elements 
of  the  area  about  any  axis  equals  the  moment  of  the  whole  area* 
about  the  same  axis,  therefore 

2# .  a  =  A .  h, 
and  P  =  w.A.h. 

16.     Centre  of  pressure. 

The  centre  of  pressure  of  any  plane 
surface  acted  upon  by  a  fluid  is  the 
point  of  action  of  the  resultant  pressure 
acting  upon  the  surface. 

Depth  of  the  centre  of  pressure.  Let 
DBG,  Fig.  9,  be  any  plane  surface 
exposed  to  fluid  pressure. 

*  See  text-books  on  Mechanics. 


14  HYDRAULICS 

Let  A  be  the  area,  and  h  the  pressure  head  at  the  centre  of 
gravity  of  the  surface,  or  if  FS  is  the  free  surface  of  the  fluid,  h  is 
the  depth  below  FS  of  the  centre  of  gravity. 
Then,  the  whole  pressure 

P  =  w.A.h. 

Let  X  be  the  depth  of  the  centre  of  pressure. 
Imagine  the  surface,  as  before,  divided  into  a  number  of  small 
areas  a,  0,1,  0$,  ...  etc. 

The  pressure  on  any  element  a 

=  w.a.x, 

and  P  =  %wax. 

Taking  moments  about  FS, 

P.X=  (wax2 


or  X- 


wAh 


~  Ah  ' 

When  the  area  is  in  a  vertical  plane,  which  intersects  the 
surface  of  the  water  in  FS,  3ax2  is  the  "second  moment"  of  the 
area  about  the  axis  FS,  or  what  is  sometimes  called  the  moment 
of  inertia  of  the  area  about  this  axis. 

Therefore,  the  depth  of  the  centre  of  pressure  of  a  vertical 
area  below  the  free  surface  of  the  fluid 

moment  of  inertia  of  the  area  about  an  axis  in  its  own  plane 
_  _  and  in  the  free  surface  _ 

area  x  the  depth  of  the  centre  of  gravity 
or,  if  I  is  the  moment  of  inertia, 


Moment  of  Inertia  about  any  axis.  Calling  I0  the  Moment 
of  Inertia  about  an  axis  through  the  centre  of  gravity,  and  I  the 
Moment  of  Inertia  about  any  axis  parallel  to  the  axis  through  the 
centre  of  gravity  and  at  a  distance  h  from  it, 

I  =  I0  +  A/>2. 

Examples.     (1)    Area  is  a  rectangle  breadth  b  and  depth  d. 

P  =  w.b.d.h, 


FLUIDS  AT   REST 


15 


If  the  free  surface  of  the  water  is  level  with  the  upper  edge  of  the  rectangle, 
^,  and  X  =  |.d. 

(2)    Area  is  a  circle  of  radius  E. 


X: 


R2 


If  the  top  of  the  circle  is  just  in  the  free  surface  or  fc= 


TABLE  II. 

Table  of  Moments  of  Inertia  of  areas. 


Form  of  area 

Moment  of  inertia  about 
an  axis  AB  through  the 
C.  of  G.  of  the  section 

Rectangle 

& 

ft™ 

Triangle 

S^f- 

r*-6-»t 

mbd3 

Circle 

J7~^B 

jfcz 

Trd4 
64 

Semicircle 

V 

About  the  axis  AB 

Trr* 
8 

Parabola 

i—  b-+* 
'*AT7*hjB 
.£W 

1* 

16 


HYDRAULICS 


17.     Diagram  of  pressure  on  a  plane  area. 

If  a  diagram  be  drawn  showing  the  intensity  of  pressure  on 
a  plane  area  at  any  depth,  the  whole  pressure  is  equal  to  the  volume 
of  the  solid  thus  formed,  and  the  centre  of  pressure  of  the  area  is 
found  by  drawing  a  line  through  the  centre 
of  gravity  of  this  solid  perpendicular  to  the 
area. 

For  a  rectangular  area  ABCD,  having  the 
side  AB  in  the  surface  of  the  water,  the 
diagram  of  pressure  is  AEFCB,  Fig.  10.  The 
volume  of  AEFCB  is  the  whole  pressure  and 
equals  \bd?w,  b  being  the  width  and  d  the 
depth  of  the  area. 

Since  the  rectangle  is  of  constant  width, 
the  diagram  of  pressure  may  be  represented 
by   the   triangle    BCF,   Fig.   11,   the    resultant   pressure   acting 
through  its  centre  of  gravity,  and  therefore  at  f  d  from  the  surface. 


h  g 

&  b  -Intensify  of  pressure 


OJV 


Fig.  12. 

-  For  a  vertical  circle  the  diagram  of  pressure  is  as  shown  in 
Figs.  12  and  13.  The  intensity  of  pressure  ab  on  any  strip  at  a 
depth  h0  is  wh^ .  The  whole  pressure  is  the  volume  of  the  truncated 
cylinder  efkh  and  the  centre  of  pressure  is  found  by  drawing  a 
line  perpendicular  to  the  circle,  through  the  centre  of  gravity 
of  this  truncated  cylinder. 


FLUIDS   AT   REST 


17 


Another,  and  frequently  a  very  convenient  method  of  deter- 
mining the  depth  of  the  centre  of  pressure,  when  the  whole  of  the 
area  is  at  some  distance  below  the  surface  of  the  water,  is  to 
consider  the  pressure  on  the  area  as  made  up  of  a  uniform  pressure 
over  the  whole  surface,  and  a  pressure  of  variable  intensity. 

Take  again,  as  an  example,  the  vertical  circle  the  diagrams  of 
pressure  for  which  are  shown  in  Figs.  12  and  13. 

At  any  depth  h  the  intensity  of  pressure  on  the  strip  ad  is 


The  pressure  on  any  strip  ad  is,  therefore,  made  up  of  a 
constant  pressure  per  unit  area  wh^  and  a  variable  pressure  whi  -f 
and  the  whole  pressure  is  equal  to  the  volume  of  the  cylinder  efgh, 
Fig.  12,  together  with  the  circular  wedge  fkg. 

The  wedge  fkg  is  equal  to  the  whole  pressure  on  a  vertical 
circle,  the  tangent  to  which  is  in  the  free  surface  of  the  water  and 

equals  w  .  A .  ^  ,  and  the  centre  of  gravity  of  this  wedge  will  be  at 

the  same  vertical  distance  from  the  centre  of  the  circle  as  the 
centre  of  pressure  when  the  circle  touches  the  surface.  The  whole 
pressure  P  may  be  supposed  therefore  to  be  the  resultant  of  two 
parallel  forces  PI  and  P2  acting  through  the  centres  of  gravity  of 
the  cylinder  efgh,  and  of  the  circular  wedge  fkg  respectively,  the 
magnitudes  of  PI  and  P2  being  the  volumes  of  the  cylinder  and 
the  wedge  respectively. 

To  find  the  centre  of  pressure  on  the  circle  AB  it  is  only 
necessary  to  find  the  resultant  of  two  parallel  forces 

of  which  Pj  acts  at  the  centre  c,  and  P2  at  a  point  Ci  which  is  at 
a  distance  from  A  of  f  r. 


Example.  A  masonry  dam,  Fig.  14, 
has  a  height  of  80  feet  from  the  founda- 
tions and  the  water  face  is  inclined  at 
10  degrees  to  the  vertical ;  find  the  whole 
pressure  on  the  face  due  to  the  water  per 
foot  width  of  the  dam,  and  the  centre  of 
pressure,  when  the  water  surface  is  level 
with  the  top  of  the  dam.  The  atmo- 
spheric pressure  may  be  neglected. 

The  whole  pressure  will  be  the  force 
tending  to  overturn  the  dam,  since  the 
horizontal  component  of  the  pressure 
on  AB  due  to  the  atmosphere  will  be 
counterbalanced  by  the  horizontal  com- 
ponents of  the  atmospheric  pressure  on 
the  back  of  the  dam.  Since  the  pressure 
on  the  face  is  normal,  and  the  intensity 
of  pressure  is  proportional  to  the  depth, 

L.  H. 


thrust 

bcLS&D  Bands  acts 
cut  th&  points  £. 

Fig.  14. 


18 


HYDRAULICS 


the  diagram  of  pressure  on  the  face  AB  will  be  the  triangle  ABC,  BC  being  equal 
to  wd  and  perpendicular  to  AB. 

The  centre  of  pressure  is  at  the  centre  of  gravity  of  the  pressure  diagram  and  is, 
therefore,  at  ^  the  height  of  the  dam  from  the  base. 

The  whole  pressure  acts  perpendicular  to  AB,  and  is  equal  to  the  area  ABC 

=  ^wd2  x  sec  10°  per  foot  width 

=  £  .  62-4  x  6400  x  1-054  =  20540  Ibs. 

Combining  P  with  W,  the  weight  of  the  dam,  the  resultant  thrust  E  on  the  base 
and  its  point  of  intersection  E  with  the  base  is  determined. 

Example.  A  vertical  flap  valve  closes  the  end  of  a  pipe  2  feet  diameter ;  the 
pressure  at  the  centre  of  the  pipe  is  equal  to  a  head  of  8  feet  of  water.  To  determine 
the  whole  pressure  on  the  valve  and  the  centre  of  pressure.  The  atmospheric 
pressure  may  be  neglected. 

The  whole  pressure  P  =  ivirR* .  8' 

=  62-4.  TT.  8  =  1570  Ibs. 

Depth  of  the  centre  of  pressure. 

The  moment  of  inertia  about  the  free  surface,  which  is  8  feet  above  the  centre 
of  the  valve,  is 


Therefore 


That  is,  |  inch  below  the  centre  of  the  valve. 

The  diagram  of  pressure  is  a  truncated  cylinder  efkh,  Figs.  12  and  13,  ef  and  hk 
being  the  intensities  of  pressure  at  the  top  and  bottom  of  the  valve  respectively. 

Example.  The  end  of  a  pontoon  which  floats  in  sea  water  is  as  shown  in  Fig.  15. 
The  level  "WL  of  the  water  is  also  shown.  Find  the  whole  pressure  on  the  end  of 
the  pontoon  and  the  centre  of  pressure. 


The  whole  pressure  on  BE 

=  64  Ibs.  x  10'  x  4-5'  x  2-25'  =  6480  Ibs. 
The  depth  of  the  centre  of  pressure  of  BE  is 

f  .4-5  =  3'. 
The  whole  pressure  on  each  of  the  rectangles  above  the  quadrants 

—  w.  5  =  320  Ibs., 
and  the  depth  of  the  centre  of  pressure  is  %  feet. 

The  two  quadrants,  since  they  are  symmetrically  placed  about  the  vertical 
centre  line,  may  be  taken  together  to  form  a  semicircle.  Let  d  be  the  distance 
below  the  centre  of  the  semicircle  of  any  element  of  area,  the  distance  of  the 
element  below  the  surface  being  h0 . 


FLUIDS   AT  REST  19 

Then  the  intensity  of  pressure  at  depth  h0 

—  w  .  2  +  w  .  d. 
And  the  whole  pressure  on  the  semicircle  is  P=w  .^-^-  .  2'  +the  whole  pressure 

on  the  semicircle  when  the  diameter  is  in  the  surface  of  the  water. 

The  distance  of  the  centre  of  gravity  of  a  semicircle  from  the  centre  of  the 
circle  is 

4R 


Therefore,  P  =  wirW  +  — 

2      Sir 

=  201R2  +  42  -66  R3  =  1256  +  666  Ibs. 

The  depth  of  the  centre  of  pressure  of  the  semicircle  when  the  surface  of  the 
water  is  at  the  centre  of  the  circle,  is 

TrR4 
8  3.7T.R 


2     '  Sir 

And,  therefore,  the  whole  pressure  on  the  semicircle  is  the  sum  of  two  forces, 
oue  of  which,  1256  Ibs.,  acts  at  the  centre  of  gravity,  or  at  a  distance  of  3-06'  from 
AD,  and  the  other  of  666  Ibs.  acts  at  a  distance  of  3-47'  from  AD. 

Then  taking  moments  about  AD  the  product  of  the  pressure  on  the  whole  area 
into  the  depth  of  the  centre  of  pressure  is  equal  to  the  moments  of  all  the  forces 
acting  on  the  area,  about  AD.  The  depth  of  the  centre  of  pressure  is,  therefore, 

6480  Ibs.  x  3'  +  320  Ibs.  x  2  x  $ '  + 1256  Ibs.  x  3-06  +  666  Ibs.  x  3-47' 

6480  +  640  + 1256  +  666 
=  2-93  feet. 


EXAMPLES. 

(1)  A  rectangular  tank  12  feet  long,  5  feet  wide,  and  5  feet  deep  is 
filled  with  water. 

Find  the  total  pressure  on  an  end  and  side  of  the  tank. 

(2)  Find  the  total  pressure  and  the  centre  of  pressure,  on  a  vertical 
sluice,  circular  in  form,  2  feet  in  diameter,  the  centre  of  which  is  4  feet 
below  the  surface  of  the  water.     [M.  S.  T.  Cambridge,  1901.] 

(3)  A  masonry  dam  vertical  on  the  water  side  supports  water  of 
120  feet  depth.     Find  the  pressure  per  square  foot  at  depths  of  20  feet  and 
70  feet  from  the  surface;  also  the  total  pressure  on  1  foot  length  of  the  dam. 

(4)  A  dock  gate  is  hinged  horizontally  at  the  bottom  and  supported  in 
a  vertical  position  by  horizontal  chains  at  the  top. 

Height  of  gate  45  feet,  width  30  ft.  Depth  of  water  at  one  side  of  the 
gate  32  feet  and  20  feet  on  the  other  side.  Find  the  tension  in  the  chains. 
Sea-water  weighs  64  pounds  per  cubic  foot. 

(5)  If  mercury  is  13£  times  as  heavy  as  water,  find  the  height  of  a 
column  corresponding  to  a  pressure  of  100  Ibs.  per  square  inch. 

(6)  A  straight  pipe  6  inches  diameter  has  a  right-angled  bend  connected 
to  it  by  bolts,  the  end  of  the  bend  being  closed  by  a  flange. 

The  pipe  contains  water  at  a  pressure  of  700  Ibs.  per  sq.  inch.  Determine 
the  total  pull  in  the  bolts  at  both  ends  of  the  elbow. 

2—2 


20 


HYDRAULICS 


(7)     The  end  of  a  dock  caisson  is  as  shown  in  Fig.  16  and  the  water 
level  is  AB. 

Determine  the  whole  pressure  and  the  centre  of  pressure. 


-----------  48'.C-"— 


—  4€.C -> 

Fig.  16. 

(8)  An  U  tube  contains  oil  having  a  specific  gravity  of  I'l  in  the  lower 
part  of  the  tube.     Above  the  oil  in  one  limb  is  one  foot  of  water,  and  above 
the  other  2  feet.     Find  the  difference  of  level  of  the  oil  in  the  two  limbs. 

(9)  A  pressure  gauge,  for  use  in  a  stokehold,  is  made  of  a  glass  U  tube 
with  enlarged  ends,  one  of  which  is  exposed  to  the  pressure  in  the  stokehold 
and  the  other  connected  to  the  outside  air.     The  gauge  is  filled  with  water 
on  one  side,  and  oil  having  a  specific  gravity  of  0*95  on  the  other — the 
surface  of  separation  being  in  the  tube  below  the  enlarged  ends.     If  the 
area  of  the  enlarged  end  is  fifty  times  that  of  the  tube,  how  many  inches  of 
water  pressure  in  the  stokehold  correspond  to  a  displacement  of  one  inch 
in  the  surface  of  separation  ?     [Lond.  Un.  1906.] 

(10)  An  inverted  oil  gauge  has  its  upper  U  filled  with  oil  having  a 
specific  gravity  of  0'7955  and  the  lower  part  of  the  gauge  is  filled  with 
water.    The  two  limbs  are  then  connected  to  two  different  points  on  a  pipe 
in  which  there  is  flowing  water. 

Find  the  difference  of  the  pressure  at  the  two  points  in  the  pipe  when 
the  difference  of  level  of  the  oil  surfaces  in  the  limbs  of  the  U  is 
15  inches. 

(11)  An  opening  in  a  reservoir  dam  is  closed  by  a  plate  3  feet  square, 
which  is  hinged  at  the  upper  horizontal  edge ;  the  plate  is  inclined  at  an 
angle  of  60°  to  the  horizontal,  and  its  top  edge  is  12  feet  below  the  surface 
of  the  water.     If  this  plate  is  opened  by  means  of  a  chain  attached  to  the 
centre  of  the  lower  edge,  find  the  necessary  pull  in  the  chain  if  its  line  of 
action  makes  an  angle  of  45°  with  the  plate.     The  weight  of  the  plate  is 
400  pounds.     [Lond.  Un.  1905.] 

(12)  The  width  of  a  lock  is  20  feet  and  it  is  closed  by  two  gates  at  each 
end,  each  gate  being  12'  long. 

If  the  gates  are  closed  and  the  water  stands  16'  above  the  bottom  on  one 
side  and  4'  on  the  other  side,  find  the  magnitude  and  position  of  the  resultant 
pressure  on  each  gate,  and  the  pressure  between  the  gates.  Show  also  that 
the  reaction  at  the  hinges  is  equal  to  the  pressure  between  the  gates.  One 
cubic  foot  of  water =62-5  Ibs.  [Lond.  Un.  1905.] 


CHAPTER  II. 

FLOATING  BODIES. 

18.     Conditions  of  equilibrium. 

When  a  body  floats  in  a  fluid  the  surface  of  the  body  in 
contact  with  the  fluid  is  subject  to  hydrostatic  pressures,  the 
intensity  of  pressure  on  any  element  of  the  surface  depend- 
ing upon  its  depth  below  the  surface.  The  resultant  of  the 
vertical  components  of  these  hydrostatic  forces  is  called  the 
buoyancy,  and  its  magnitude  must  be  exactly  equal  to  the  weight 
of  the  body,  for  if  not  the  body  will  either  rise  or  sink.  Again 
the  horizontal  components  of  these  hydrostatic  forces  must 
be  in  equilibrium  amongst  themselves,  otherwise  the  body  will 
have  a  lateral  movement. 

The  position  of  equilibrium  for  a  floating  body  is  obtained 
when  (a)  the  buoyancy  is  exactly  equal  to  the  weight  of  the 
body,  and  (6)  the  vertical  forces — the  weight  and  the  buoyancy- 
act  in  the  same  vertical  line,  or  in  other  words,  in  such  a  way  as 
to  produce  no  couple  tending  to  make  the  body  rotate. 

Let  Gr,  Fig.  17,  be  the  centre  of  gravity  of  a  floating  ship  and 
BK,  which  does  not  pass  through  Gr,  the  line  of  action  of  the 
resultant  of  the  vertical  buoyancy  forces.  Since  the  buoyancy 


Fig.  17. 


Fig.  18. 


must  equal  the  weight  of  the  ship,  there  are  two  parallel  forces 
each  equal  to  W  acting  through  Gr  and  along  BK  respectively, 
and  these  form  a  couple  of  magnitude  Wa?,  which  tends  to  bring 
the  ship  into  the  position  shown  in  Fig.  18,  that  is,  so  that  BK 


22 


HYDRAULICS 


passes  through  Gr.  The  above  condition  (6)  can  therefore  only  be 
realised,  when  the  resultant  of  the  buoyancy  forces  passes  through 
the  centre  of  gravity  of  the  body.  If,  however,  the  body  is 
displaced  from  this  position  of  equilibrium,  as  for  example  a  ship 
at  sea  would  be  when  made  to  roll  by  wave  motions,  there  will 
generally  be  a  couple,  as  in  Fig.  17,  acting  upon  the  body,  which 
should  in  all  cases  tend  to  restore  the  body  to  its  position  of 
equilibrium.  Consequently  the  floating  body  will  oscillate  about 
its  equilibrium  position  and  it  is  then  said  to  be  in  stable  equi- 
librium. On  the  other  hand,  if  when  the  body  is  given  a  small 
displacement  from  the  position  of  equilibrium,  the  vertical  forces 
act  in  such  a  way  as  to  cause  a  couple  tending  to  increase  the 
displacement,  the  equilibrium  is  said  to  be  unstable. 

The  problems  connected  with  floating  bodies  acted  upon  by 
forces  due  to  gravity  and  the  hydrostatic  pressures  only, 
resolve  themselves  therefore  into  two, 

(a)     To  find  the  position  of  equilibrium  of  the  body. 

(6)     To  find  whether  the  equilibrium  is  stable. 

19.    Principle  of  Archimedes. 

When  a  body  floats  freely  in  a  fluid  the  weight  of  the  body  is 
equal  to  the  weight  of  the  fluid  displaced. 

Since  the  weight  of  the  body  is  equal  to  the  resultant  of  the 
vertical  hydrostatic  pressures,  or  to  the  buoyancy,  this  principle 
will  be  proved,  if  the  weight  of  the  water  displaced  is  shown  to  be 
equal  to  the  buoyancy. 

Let  ABC,  Fig.  19,  be  a  body  floating  in  equilibrium,  AC  being 
in  the  surface  of  the  fluid. 


Fig.  19. 

Consider  any  small  element  ab  of  the  surface,  of  area  a  and 
depth  h,  the  plane  of  the  element  being  inclined  at  any  angle  0  to 
the  horizontal.  Then,  if  w  is  the  weight  of  unit  volume  of  the 
fluid,  the  whole  pressure  on  the  area  a  is  wha,  and  the  vertical 
component  of  this  pressure  is  seen  to  be  wha  cos  0. 


FLOATING   BODIES  23 

Imagine  now  a  vertical  cylinder  standing  on  this  area,  the  top 
of  which  is  in  the  surface  AC. 

The  horizontal  sectional  area  of  this  cylinder  is  a  cos  0,  the 
volume  is  ha  cos  0  and  the  weight  of  the  water  filling  this  volume 
is  whacosO,  and  is,  therefore,  equal  to  the  buoyancy  on  the 
area  ab. 

If  similar  cylinders  be  imagined  on  all  the  little  elements 
of  area  which  make  up  the  whole  immersed  surface,  the  total 
volume  of  these  cylinders  is  the  volume  of  the  water  displaced, 
and  the  total  buoyancy  is,  therefore,  the  weight  of  this  displaced 
water. 

If  the  body  is  wholly  immersed  as  in  Fig.  20  and  the 
body  is  supposed  to  be  made  up  of  small  .  _ 

vertical  cylinders  intersecting  the  surface  of 
the  body  in  the  elements  of  area  ab  and  ab', 
which  are  inclined  to  the  horizontal  at  angles 
0  and  <£  and  having  areas  a  and  ai  respectively, 
the  vertical  component  of  the  pressure  on  ab 
will  be  wha cos 0  and  on  ab'  will  be  whidi cos <£. 
But  acos#  must  equal  o^cos^,  each  being 
equal  to  the  horizontal  section  of  the  small  cylinder.  The  whole 
buoyancy  is  therefore 

2>wha  cos  0  —  ^whiai  cos  <£, 
and  is  again  equal  to  the  weight  of  the  water  displaced. 

In  this  case  if  the  fluid  be  assumed  to  be  of  constant  density 
and  the  weight  of  the  body  as  equal  to  the  weight  of  the  fluid 
of  the  same  volume,  the  body  will  float  at  any  depth.  The 
slightest  increase  in  the  weight  of  the  body  would  cause  it  to 
sink  until  it  reached  the  bottom  of  the  vessel  containing  the  fluid, 
while  a  very  small  diminution  of  its  weight  or  increase  in  its 
volume  would  cause  it  to  rise  immediately  to  the  surface.  It 
would  clearly  be  practically  impossible  to  maintain  such  a  body 
in  equilibrium,  by  endeavouring  to  adjust  the  weight  of  the  body, 
by  pumping  out,  or  letting  in,  water,  as  has  been  attempted  in  a 
certain  type  of  submarine  boat.  In  recent  submarines  the  lowering 
and  raising  of  the  boat  are  controlled  by  vertical  screw  propellers. 

20.     Centre  of  buoyancy. 

Since  the  buoyancy  on  any  element  of  area  is  the  weight  of 
the  vertical  cylinder  of  the  fluid  above  this  area,  and  that  the 
whole  buoyancy  is  the  sum  of  the  weights  of  all  these  cylinders,  it 
at  once  follows,  that  the  resultant  of  the  buoyancy  forces  must 
pass  through  the  centre  of  gravity  of  the  water  displaced,  and  this 
point  is,  therefore,  called  the  Centre  of  Buoyancy. 


•24 


HYDRAULICS 


21.     Condition  of  stability  of  equilibrium. 

Let  AND,  Fig.  21,  be  the  section  made  by  a  vertical  plane 
containing  Gr  the  centre  of  gravity  and  B  the  centre  of  buoyancy 
of  a  floating  vessel,  AD  being  the  surface  of  the  fluid  when  the 
centre  of  gravity  and  centre  of  buoyancy  are  in  the  same  vertical 
line. 


B 


Fig.  21. 


M, 

B, 

Fig.  22. 


Let  the  vessel  be  heeled  over  about  a  horizonal  axis,  FE  being 
now  the  fluid  surface,  and  let  Bx  be  the  new  centre  of  buoyancy, 
the  above  vertical  sectional  plane  being  taken  to  contain  Gr,  B, 
and  BI.  Draw  BiM,  the  vertical  through  B1}  intersecting  the  line 
GrB  in  M.  Then,  if  M  is  above  Gr  the  couple  W .  x  will  tend  to 
restore  the  ship  to  its  original  position  of  equilibrium,  but  if  M  is 
below  Gr,  as  in  Fig.  22,  the  couple  will  tend  to  cause  a  further 
displacement,  and  the  ship  will  either  topple  over,  or  will  heel  over 
into  a  new  position  of  equilibrium. 

In  designing  ships  it  is  necessary  that,  for  even  large  displace- 
ments such  as  may  be  caused  by  the  rolling  of  the  vessel,  the 
point  M  shall  be  above  Gr.  To  determine  M,  it  is  necessary  to 
determine  Gr  and  the  centres  of  buoyancy  for  the  two  positions 
of  the  floating  body.  This  in  many  cases  is  a  long  and  somewhat 
tedious  operation. 

22.     Small  displacements.     Metacentre. 

When  the  angular  displacement  is  small  the  point  M  is  called 
the  Metacentre,  and  the  distance  of  M  from  Gr  can  be  calculated. 

Assume  the  angular  displacement  in  Fig.  21  to  be  small  and 
equal  to  0. 

Then,  since  the  volume  displacement  is  constant  the  volume  of 
the  wedge  ODE  must  equal  CAF,  or  in  Fig.  23,  dC2DE  must  equal 
dC2AF. 


FLOATING   BODIES 


25 


Let  Gri  and  Gr2  be  the  centres  of  gravity  of  the  wedges  CiC2AF 
and  CiC2DE  respectively. 


d-f    D 

Fig.  23. 

The  heeling  of  the  ship  has  the  effect  of  moving  a  mass  of 
water  equal  to  either  of  these  wedges  from  GK  to  Gr2,  and  this 
movement  causes  the  centre  of  gravity  of  the  whole  water 
displaced  to  move  from  B  to  Ba . 

Let  Z  be  the  horizontal  distance  between  (TI  and  G2,  when  FE 
is  horizontal,  and  S  the  perpendicular  distance  from  B  to  BiM. 

Let  V  be  the  total  volume  displacement,  v  the  volume  of  the 
wedge  and  w  the  weight  of  unit  volume  of  the  fluid. 

Then  w  .  v .  Z  =  w  .  Y .  S 

=  ™.V.BM.sin0. 

Or,  since  0  is  small,  =  w.V.BM.0  (1). 

The  restoring  couple  is 

•  =  w.V.G-M.0 


rr  ~\T     ~DC*     A  /O^ 

=  IV  •  V  •  £J  —  10  .  V  .  JjljT  .  "     \£)  • 

But  w  .  v .  Z  =  twice  the  sum  of  the  moments  about  the  axis 
of  all  the  elements  such  as  acdb  which  make  up  the  wedge 
CADE. 

Taking  ab  as  x,  bf  is  x&9  and  if  ac  is  dl,  the  volume  of  the 
element  is  Ja?20 .  dl. 

The  centre  of  gravity  of  the  element  is  at  fa?  from  CiC2. 

Therefore  to .  t> .  Z  =  2iflfl  -^-    (3). 


,  is  the  Second  Moment  or  Moment  of  Inertia  of  the 

o 

element  of  area  aceb  about  C2Ci,  and  2  J    -g-  is,  therefore,  the 
Moment  of  Inertia  I  of  the  water-plane  area  ACiDCa  about  CiC2. 
Therefore  w.v.Z  =  w.I.O (4). 


26  HYDRAULICS 

The  restoring  couple  is  then 


If  this  is  positive,  the  equilibrium  is  stable,  but  if  negative  it  is 
unstable. 

Again  since  from  (1) 

wv  .  Z  -  w  .  Y  .  BM  .  0, 
therefore  w  .  Y  .  BM  .  6  =  wI0, 


and 


If  BM  is  greater  than  BGr  the  equilibrium  is  stable,  if  less  than 
BGr  it  is  unstable,  and  the  body  will  heel  over  until  a  new  position 
of  equilibrium  is  reached.  If  BGr  is  equal  to  BM  the  equilibrium 
is  said  to  be  neutral. 

The  distance  G-M  is  called  the  Metacentric  Height,  and  varies 
in  various  classes  of  ships  from  a  small  negative  value  to  a  positive 
value  of  4  or  5  feet. 

When  the  metacentric  height  is  negative  the  ship  heels  until 
it  finds  a  position  of  stable  equilibrium.  This  heeling  can  be 
corrected  by  ballasting. 

Example.     A  ship  has  a  displacement  of  15,400  tons,  and  a  draught  of  27*5  feet. 
The  height  of  the  centre  of  buoyancy  from  the  bottom  of  the  keel  is  15  feet. 

The  moment  of  inertia  of  the  horizontal  section  of  the  ship  at  the  water  line 
is  9,400,000  feet4  units. 

Determine  the  position  of  the  centre  of  gravity  that  the  metacentric  height  shall 
not  be  less  than  4  feet  in  sea  water. 

9,400,000x64 
"15,400x2240 
=  17-1  feet. 

Height  of  metacentre  from  the  bottom  of  the  keel  is,  therefore,  32*1  feet. 
As  long  as  the  centre  of  gravity  is  not  higher  than  0'6  feet  above  the  surface  of 
the  water,  the  metacentric  height  is  more  than  4  feet. 

23.     Stability  of  a  rectangular  pontoon. 

Let  EFJS,  Fig.  24,  be  the  section  of  the  pontoon  and  Gr  its 
centre  of  gravity. 

Let  YE  be  the  surface  of  the  water  when  the  sides  of  the 
pontoon  are  vertical,  and  AL  the  surface  of  the  water  when  the 
pontoon  is  given  an  angle  of  heel  0. 

Then,  since  the  weight  of  water  displaced  equals  the  weight  of 
the  pontoon,  the  area  AFJL  is  equal  to  the  area  YFJE. 

Let  B  be  the  centre  of  buoyancy  for  the  vertical  position, 
B  being  the  centre  of  area  of  VF  JE,  and  BI  the  centre  of  buoyancy 
for  the  new  position,  BI*  being  the  centre  of  area  of  AFJL.  Then 
the  line  joining  BGr  must  be  perpendicular  to  the  surface  YE  and 

*  In  the  Fig.,  Bt  is  not  the  centre  of  area  of  APJL,  as,  for  the  sake  of  clearness, 
it  is  further  removed  from  B  than  it  actually  should  be. 


FLOATING  BODIES 


27 


is  the  direction  in  which  the  buoyancy  force  acts  when  the  sides 
of  the  pontoon  are  vertical,  and  BjM  perpendicular  to  AL  is  the 
direction  in  which  the  buoyancy  force  acts  when  the  pontoon  is 
heeled  over  through  the  angle  0.  M  is  the  metacentre. 


F 


Fig.  24. 


The  forces  acting  on  the  pontoon  in  its  new  position  are,  W  the 
weight  of  the  pontoon  acting  vertically  through  G  and  an  equal  and 
parallel  buoyancy  force  W  through  BI  . 

There  is,  therefore,  a  couple,  W.HGr,  tending  to  restore  the 
pontoon  to  its  vertical  position. 

If  the  line  BiH  were  to  the  right  of  the  vertical  through  Gr,  or 
in  other  words  the  point  M  was  below  Gr,  the  pontoon  would  be  in 
unstable  equilibrium. 

The  new  centre  of  buoyancy  Ba  can  be  found  in  several  ways. 
The  following  is  probably  the  simplest. 

The  figure  AFJL  is  formed  by  moving  the  triangle,  or  really 
the  wedge-shaped  piece  GEL  to  CYA,  and  therefore  it  may  be 
imagined  that  a  volume  of  water  equal  to  the  volume  of  this  wedge 
is  moved  from  Gr2  to  GK .  This  will  cause  the  centre  of  buoyancy 
to  move  parallel  to  GriGr2  to  a  new  position  BI,  such  that 

BBi  x  weight  of  pontoon  =  GiGa  x  weight  of  water  in  GEL. 

Let  b  be  half  the  breadth  of  the  pontoon, 
Z  the  length, 

D  the  depth  of  displacement  for  the  upright  position, 
d  the  length  LE,  or  AY, 
and      w  the  weight  of  a  cubic  foot  of  water. 

Then,  the  weight  of  the  pontoon 

bd 
and  the  weight  of  the  wedge  CLE  =  -^  x  I  •  w. 


28  HYDRAULICS 

Therefore  BBi .  26 .  D  =  G^b'd , 

d  rr 
and  -D±>i  =  ^g  Irilr2. 

Resolving  BBi  and  GriGr2,  which  are  parallel  to  each  other,  along 
and  perpendicular  to  BM  respectively, 

_d_  J_/2     \      Id  ^VtanO 


To  find  the  distance  of  the  point  M  from  G  and  the  value  of  the 
restoring  couple.  Since  BiM  is  perpendicular  to  AL  and  BM  to 
YE,  the  angle  BMBi  equals  6. 

Therefore  QM  =  BXQ  cot  6  =  |g  cot  0  =  j^  . 

Let  z  be  the  distance  of  the  centre  of  gravity  Gr  from  C. 
Then  Qa  =  QC-z  =  BC-BQ-z 

D     &2tan20 


~  2         6D 
Therefore 

&        ^       fr2  tan2  0 


And  since  HGr  =  GrM  sin  0, 

the  righting  couple, 


The  distance  of  the  metacentre  from  the  point  B,  is 
QM  +  QB^Qcottf  +  ^j^ 

V_      62tan20 
~  3D  +      6D 
When  0  is  small,  the  term  containing  tan2  6  is  negligible,  and 


This    result    can    be    obtained    from    formula    (4)    given    in 
section  22. 

I  for  the  rectangle  is  TV  (26)3  =  f  Z63,  and  V  =  26DZ. 

Therefore  BM=Ji. 

<L>U 

If  BGr  is  known,  the  metacentric  height  can  now  be  found. 


FLOATING   BODIES 


29 


Example.  A  pontoon  has  a  displacement  of  200  tons.  Its  length  is  50  feet. 
The  centre  of  gravity  is  1  foot  above  the  centre  of  area  of  the  cross  section.  Find 
the  breadth  and  depth  of  the  pontoon  so  that  for  an  angular  displacement  of  10  degrees 
the  metacentre  shall  not  be  less  than  3  feet  from  the  centre  of  gravity,  and  the  free- 
board shall  not  be  less  than  2  feet. 

Kef  erring  to  Fig.  24,  G  is  the  centre  of  gravity  of  the  pontoon  and  0  is  the 
centre  of  the  cross  section  KJ. 

Then,  GO  =  1  foot, 

F0  =  2feet, 

GM  =  3  feet. 

Let  D  be  the  depth  of  displacement.     Then 

D  x  26  x  62-4  x  50  Ibs.  =  200  tons  x  2240  Ibs. 
Therefore  D6  =  71'5 


(1). 


The  height  of  the  centre  of  buoyancy  B  above  the  bottom  of  pontoon  is 


Since  the  free-board  is  to  be  2  feet, 


Then 

Therefore 

But 


=  l'  and  BG  =  2'. 
BM  =  5'. 


tan2  0 


6D 


(2). 


Multiplying  numerator  and  denominator  by  b,  and  substituting  from  equation  (1) 


214-5 


from  which 

therefore 

and 


429 


»-^  6  =  10-1  ft., 

D  =  7'lft., 

The  breadth  B  =  20-2  ft.l 
,,     depth 


!  =  20-2  ft.) 
=   7-1  ft.) 


Ans. 


24.     Stability  of  a  floating  vessel  containing  water. 

If  a  vessel  contains  water  with  a  free  surface,  as  for  instance 
the  compartments  of  a  floating  dock,  such  as  is  described  on  page 
31,  the  surface  of  the  water  in  these  compartments  will  remain 
horizontal  as  the  vessel  heels  over,  and  the  centre  of  gravity  of 
the  water  in  any  compartment  will  change  its  position  in  such 
a  way  as  to  increase  the  angular  displacement  of  the  vessel. 

In  considering  the  stability 
of  such  vessels,  therefore,  the 
turning  moments  due  to  the 
water  in  the  vessel  must  be 
taken  into  account. 

As  a  simple  case  consider 
the  rectangular  vessel,  Fig.  25, 
which,  when  its  axis  is  vertical, 
floats  with  the  plane  AB  in  the  Fig-  25' 


A 

_---~ 
E 
H 

_____ 

_„.,-—  '-"" 

.—  — 
11 

K 
J) 

__^ 

^' 

>^-A 

30 


HYDRAULICS 


surface  of  the  fluid,  DE  being  the  surface  of  the  fluid  in  the 
vessel. 

When  the  vessel  is  heeled  through  an  angle  0,  the  surface  of 
fluid  in  the  vessel  is  KH. 

The  effect  has  been,  therefore,  to  move  the  wedge  of  fluid  OEH 
to  ODK,  and  the  turning  couple  due  to  this  movement  is  w  .  v  .  Z, 
v  being  the  volume  of  either  wedge  and  Z  the  distance  between 
the  centre  of  gravity  of  the  wedges. 

62 
If  26  is  the  width  of  the  vessel  and  I  its  length,  v  is  -~  I  tan  0, 

Z  is  ^b  tan  0,  and  the  turning  couple  is  w  f  63 1  tan2  0. 

If  0  is  small  wvZ  is  equal  to  wW,  I  being  the  moment  of  inertia 
of  the  water  surface  KH  about  an  axis  through  0,  as  shown  in 
section  22. 

For  the  same  width  and  length  of  water  surface  in  the 
compartment,  the  turning  couple  is  the  same  wherever  the 
compartment  is  situated,  for  the  centre  of  gravity  of  the  wedge 
OHE,  Fig.  26,  is  moved  by  the  same  amount  in  all  cases. 

If,  therefore,  there  are  free  fluid  surfaces  in  the  floating  vessel, 
for  any  small  angle  of  heel  0,  the  tippling-moment  due  to  these 
surfaces  is  2wI0,  I  being  in  all  cases  the  moment  of  inertia  of  the 
fluid  surface  about  its  own  axis  of  oscillation,  or  the  axis  through 
the  centre  of  gravity  of  the  surface. 


H 


0 


Fig.  26. 


Fig.  27. 


25.     Stability  of  a  floating  body  wholly  immersed. 

It  has  already  been  shown  that  a  floating  body  wholly  im- 
mersed in  a  fluid,  as  far  as  vertical  motions  are  concerned,  can 
only  with  great  difficulty  be  maintained  in  equilibrium. 

If  further  the  body  is  made  to  roll  through  a  small  angle,  the 
equilibrium  will  be  unstable  unless  the  centre  of  gravity  of  the 
body  is  below  the  centre  of  buoyancy.  This  will  be  seen  at  once 
on  reference  to  Fig.  27.  Since  the  body  is  wholly  immersed  the 
centre  of  buoyancy  cannot  change  its  position  on  the  body  itself, 
as  however  it  rolls  the  centre  of  buoyancy  must  be  the  centre  of 
gravity  of  the  displaced  water,  and  this  is  not  altered  in  form  by 


FLOATING  BODIES 


31 


any  movement  of  the  body.  If,  therefore,  Gr  is  above  B  and  the 
body  be  given  a  small  angular  displacement  to  the  right  say,  Gr 
will  move  to  the  right  relative  to  B  and  the  couple  will  not  restore 
the  body  to  its  position  of  equilibrium. 

On  the  other  hand,  if  Gr  is  below  B,  the  couple  will  act  so  as  to 
bring  the  body  to  its  position  of  equilibrium. 

26.     Floating  docks. 

Figs.  28  and  29  show  a  diagrammatic  outline  of  the  pontoons 
forming  a  floating  dock,  and  in  the  section  is  shown  the  outline  of 
a  ship  on  the  dock. 


w 


L. 


Fig.  29. 

To  dock  a  ship,  the  dock  is  sunk  to  a  sufficient  depth  by 
admitting  water  into  compartments  formed  in  the  pontoons,  and  the 
ship  is  brought  into  position  over  the  centre  of  the  dock. 

Water  is  then  pumped  from  the  pontoon  chambers,  and  the 
dock  in  consequence  rises  until  the  ship  just  rests  on  the  keel 
blocks  of  the  dock.  As  more  water  is  pumped  from  the  pontoons 
the  dock  rises  with  the  ship,  which  may  thus  be  lifted  clear  of 
the  water. 

Let  Gri  be  the  centre  of  gravity  of  the  ship,  Gr2  of  the  dock  and  i 
water  ballast  and  Gr  the  centre  of  gravity  of  the  dock  and  the 


ship. 


The  position  of  the  centre  of  gravity  of  the  dock  will  vary 


32  HYDRAULICS 

relative  to  the  bottom  of  the  dock,  as  water  is  pumped  from  the 
pontoons. 

As  the  dock  is  raised  care  must  be  taken  that  the  metacentre 
is  above  Gr  or  the  dock  will  "  list." 

Suppose  the  ship  and  dock  are  rising  and  that  WL  is  the 
water  line. 

Let  B2  be  the  centre  of  buoyancy  of  the  dock  and  B:  of  the 
portion  of  the  ship  still  below  the  water  line. 

Then  if  Vi  and  V2  are  the  volume  displacements  below 
the  water  line  of  the  ship  and  dock  respectively,  the  centre  of 
buoyancy  B  of  the  whole  water  displaced  divides  B2B3,  so  that 


BB2~Y/ 

The  centre  of  gravity  Gr  of  the  dock  and  the  ship  divides  GriGr2 
in  the  inverse  ratios  of  their  weights. 

As  the  dock  rises  the  centre  of  gravity  Gr  of  the  dock  and  the 
ship  must  be  on  the  vertical  through  B,  and  water  must  be 
pumped  from  the  pontoons  so  as  to  fulfil  this  condition  and  as 
nearly  as  possible  to  keep  the  deck  of  the  dock  horizontal. 

The  centre  of  gravity  Gri  of  the  ship  is  fixed,  while  the  centre  of 
buoyancy  of  the  ship  BI  changes  its  position  as  the  ship  is  raised. 

The  centre  of  buoyancy  B2  of  the  dock  will  also  be  changing, 
but  as  the  submerged  part  of  the  dock  is  symmetrical  about  its 
centre  lines,  B2  will  only  move  vertically.  As  stated  above,  B 
must  always  lie  on  the  line  joining  BI  and  B2,  and  as  Gr  is  to  be 
vertically  above  B,  the  centre  of  gravity  Gr2  and  the  weight  of 
the  pontoon  must  be  altered  by  taking  water  from  the  various 
compartments  in  such  a  way  as  to  fulfil  this  condition. 

Quantity  of  water  to  be  pumped  from  the  pontoons  in  raising  the 
dock.  Let  V  be  the  volume  displacement  of  the  dock  in  its  lowest 
position,  Yo  the  volume  displacement  in  its  highest  position.  To 
raise  the  dock  without  a  ship  in  it  the  volume  of  the  water  to  be 
pumped  from  the  pontoons  is  Y  -  Y0  . 

If,  when  the  dock  is  in  its  highest  position,  a  weight  W  is  put 
on  to  the  dock,  the  dock  will  sink,  and  a  further  volume  of  water 

W 

-  cubic  feet  will  be  required  to  be  taken  from  the  pontoons  to 

raise  the  dock  again  to  its  highest  position. 

To  raise  the  dock,  therefore,  and  the  ship,  a  total  quantity  of 
water 


cubic  feet  will  have  to  be  taken  from  the  pontoons. 


FLOATING   BODIES  33 

Example.  A  floating  dock  as  shown  dimensioned  in  Fig.  28  is  made  up  of  a 
bottom  pontoon  540  feet  long  x  96  feet  wide  x  14-75  feet  deep,  two  side  pontoons 
380  feet  long  x  13  feet  wide  x  4-8  feet  deep,  the  bottom  of  these  pontoons  being 
2  feet  above  the  bottom  of  the  dock,  and  two  side  chambers  on  the  top  of  the 
bottom  pontoon  447  feet  long  by  8  feet  deep  and  2  feet  wide  at  the  top  and  8  feet  at 
the  bottom.  The  keel  blocks  may  be  taken  as  4  feet  deep. 

The  dock  is  to  lift  a  ship  of  15,400  tons  displacement  and  27'  6"  draught. 

Determine  the  amount  of  water  that  must  be  pumped  from  the  dock,  to  raise 
the  ship  so  that  the  deck  of  the  lowest  pontoon  is  in  the  water  surface. 

When  the  ship  just  takes  to  the  keel  blocks  on  the  dock,  the  bottom  of  the 
dock  is  27-5' +  14-75' +  4' =  46 -25  feet  below  the  water  line. 

The  volume  displacement  of  the  dock  is  then 

14-75  x  540  x  96  +  2  x  44-25  x  13  x  380  +  447  x  8  x  5' =  1,237, 600  cubic  feet. 
The  volume  of  dock  displacement  when  the  deck  is  just  awash  is 

540  x  96  x  14-75  +  2  x  380  x  13'  x  (14-75  -  2)  =  890,000  cubic  feet. 
The  volume  displacement  of  the  ship  is 
15,400  x  2240 


64 


=  540,000  cubic  feet, 


and  this  equals  the  weight  of  the  ship  in  cubic  feet. 

Of  the  891,000  cubic  feet  displacement  when  the  ship  is  clear  of  the  water, 
351,000  cubic  feet  is  therefore  required  to  support  the  dock  alone. 

Simply  to  raise  the  dock  through  31 '5  feet  the  amount  of  water  to  be  pumped  is 
the  difference  of  the  displacements,  and  is,  therefore,  347,600  cubic  feet. 

To  raise  the  ship  with  the  dock  an  additional  540,000  cubic  feet  must  be 
extracted  from  the  pontoons. 

The  total  quantity,  therefore,  to  be  taken  from  the  pontoons  from  the  time  the 
ship  takes  to  the  keel  blocks  to  when  the  pontoon  deck  is  in  the  surface  of  the 
water  is 

887,600  cubic  feet  =  25,380  tons. 

27.     Stability  of  the  floating  dock. 

As  some  of  the  compartments  of  the  dock  are  partially  filled 
with  water,  it  is  necessary,  in  considering  the  stability,  to  take 
account  of  the  tippling-moments  caused  by  the  movement  of  the 
free  surface  of  the  water  in  these  compartments. 

If  Gr  is  the  centre  of  gravity  of  the  dock  and  ship  on  the 
dock,  B  the  centre  of  buoyancy,  I  the  moment  of  inertia  of  the 
section  of  the  ship  and  dock  by  the  water-plane  about  the  axis  of 
oscillation,  and  I1?  I2  etc.  the  moments  of  inertia  of  the  water 
surfaces  in  the  compartments  about  their  axes  of  oscillation,  the 
righting  moment  when  the  dock  receives  a  small  angle  of 
heel  0,  is 

wIO  -  ™  (Va  +  Y2)  RGO-wO  (Ix  + 12  +...). 

The  moment  of  inertia  of  the  water-plane  section  varies 
considerably  as  the  dock  is  raised,  and  the  stability  varies 
accordingly. 

When  the  ship  is  immersed  in  the  water,  I  is  equal  to  the 
moment  of  inertia  of  the  horizontal  section  of  the  ship  at  the 
water  surface,  together  with  the  moment  of  inertia  of  the 
horizontal  section  of  the  side  pontoons,  about  the  axis  of 
oscillation  0. 

L.H.  3 


34  HYDRAULICS 

When  the  tops  of  the  keel  blocks  are  just  above  the  surface 
of  the  water,  the  water-plane  is  only  that  of  the  side  pontoons, 
and  I  has  its  minimum  value.     If  the  dock  is  L-shaped  as  in 
Fig.  30,  which  is  a  very  convenient  form 
for  some  purposes,  the   stability  when 
the  tops  of  the  keel  blocks  come  to  the 
surface  simply  depends  upon  the  moment 
of  inertia  of  the  area  AB  about  an  axis   : 
through  the  centre  of  AB.    This  critical 
point   can,  however,  be   eliminated  by 


fitting  an  air  box,  shown  dotted,  on  the  Fi     30 

outer  end  of  the  bottom  pontoon,  the 

top  of  which  is  slightly  higher  than  the  top  of  the  keel  blocks. 

Example.  To  find  the  height  of  the  metacentre  above  the  centre  of  buoyancy  of 
the  dock  of  Fig.  28  when 

(a)     the  ship  just  takes  to  the  keel  blocks, 
(6)     the  keel  is  just  clear  of  the  water, 
(c)     the  pontoon  deck  is  just  above  the  water. 

Take  the  moment  of  inertia  of  the  horizontal  section  of  the  ship  at  the 
water  line  as  9,400,000  ft.4  units,  and  assume  that  the  ship  is  symmetrically 
placed  on  the  dock,  and  that  the  dock  deck  is  horizontal.  The  horizontal  distance 
between  the  centres  of  the  side  tanks  is  111  ft. 

(a)     Total  moment  of  inertia  of  the  horizontal  section  is 

9,  400,000  +  2  (380  x  13'  x  55  "52  +  T\  x  380  x  133)  =  9,400,000  +  30,430,000  +  139,000. 
The  volume  of  displacement 

=  540,000  +  1,237,600  cubic  feet. 
The  height  of  the  metacentre  above  the  centre  of  buoyancy  is  therefore 


(b)  When  the  keel  is  just   clear   of    the  water   the    moment   of    inertia    is 
30,569,000. 

The  volume  displacement  is 

540  x  96  x  14-75  +  380  x  2  x  13  x  (14-75  +  4-2) 

=  930,000  cubic  feet. 
Therefore  BM  =  32-8  feet. 

(c)  When  the  pontoon  deck  is  just  above  the  surface  of  the  water, 

I  =  30,569,000  +  ~TV  x  510'  x  963 

=  70,269,000. 

The  volume  displacement  is  890,000  cubic  feet. 
Therefore  BM  =  79'8  feet. 

The  height  of  the  centre  of  buoyancy  above  the  bottom  of  the  dock  can  be 
determined  by  finding  the  centre  of  buoyancy  of  each  of  the  parts  of  the  dock,  and 
of  the  ship  if  it  is  in  the  water,  and  then  taking  moments  about  any  axis. 

For  example.  To  find  the  height  h  of  the  centre  of  buoyancy  of  the  dock  and 
the  ship,  when  the  ship  just  comes  on  the  keel  blocks. 

The  centre  of  buoyancy  for  the  ship  is  at  15  feet  above  the  bottom  of  the  keel. 
The  centre  of  buoyancy  of  the  bottom  pontoon  is  at     7  -375'  from  the  bottom. 
»  »  5»  ,,        side  pontoons         ,,     24-125'       ,,  ,, 

»»  »  »>  „         ,,     chambers        ,,     17'94' 


FLOATING   BODIES  35 

Taking  moments  about  the  bottom  of  the  dock 

h  (540,000  +  437,000  +  765,000  +  35,760) 
=  540,000  x  33-75  +  765,000  x  7-375 
+  437,000  x  24-125  +  35, 760  x  17  "95, 
therefore  h  =  19  '1  feet. 

For  case  (a)  the  metacentre  is,  therefore,  40-3'  above  the  bottom  of  the  dock.    If 
now  the  centre  of  gravity  of  the  dock  and  ship  is  known  the  metacentric  height 


can  be  found. 


EXAMPLES. 


(1)  A  ship  when  fully  loaded  has  a  total  burden  of  10,000  tons.     Find 
the  volume  displacement  in  sea  water. 

(2)  The  sides  of  a  ship  are  vertical  near  the  water  line  and  the  area  of 
the  horizontal  section  at  the  water  line  is  22,000  sq.  feet.    The  total  weight 
of  the  ship  is  10,000  tons  when  it  leaves  the  river  dock. 

Find  the  difference  in  draught  in  the  dock  and  at  sea  after  the  weight 
of  the  ship  has  been  reduced  by  consumption  of  coal,  etc.,  by  1500  tons. 
Let  9  be  the  difference  in  draught. 
Then  d  x  22,000  =  the  difference  in  volume  displacement 
_  10,000  x  2240     8500  x  2240 

62-43  64 

=6130  cubic  feet. 
Therefore        8  =  '278  feet 

=3'34  inches. 

(3)  The  moment  of  inertia  of  the  section  at  the  water  line  of  a  boat 
is  1200  foot4  units;   the  weight  of  the  boat  is  11-5  tons. 

Determine  the  height  of  the  metacentre  above  the  centre  of  buoyancy. 

(4)  A  ship  has  a  total  displacement  of  15.000  tons  and  a  draught  of 
27  feet. 

When  the  ship  is  lifted  by  a  floating  dock  so  that  the  depth  of  the  bottom 
of  the  keel  is  16'5  feet,  the  centre  of  buoyancy  is  10  feet  from  the  bottom  of 
the  keel  and  the  displacement  is  9000  tons. 

The  moment  of  inertia  of  the  water-plane  is  7,600,000  foot4  units. 

The  horizontal  section  of  the  dock,  at  the  plane  16'5  feet  above  the 
bottom  of  the  keel,  consists  of  two  rectangles  380  feet  x  11  feet,  the  distance 
apart  of  the  centre  lines  of  the  rectangles  being  114  feet. 

The  volume  displacement  of  the  dock  at  this  level  is  1,244,000  cubic  feet. 

The  centre  of  buoyancy  for  the  dock  alone  is  24'75  feet  below  the  surface 
of  the  water. 

Determine  (a)  The  centre  of  buoyancy  for  the  whole  ship  and  the  dock. 

(b)  The  height  of  the  metacentre  above  the  centre  of  buoyancy. 

(5)  A  rectangular  pontoon  60  feet  long  is  to  have  a  displacement  of 
220  tons,  a  free-board  of  not  less  than  3  feet,  and  the  metacentre  is  not  to 
be  less  than  3  feet  above  the  centre  of  gravity  when  the  angle  of  heel 
is  15  degrees.     The  centre  of  gravity  coincides  with  the  centre  of  figure. 

Find  the  width  and  depth  of  the  pontoon. 

3—2 


36  HYDRAULICS 

(6)  A  rectangular  pontoon  24  feet  wide,  50  feet  long  and  14  feet  deep, 
has  a  displacement  of  180  tons. 

A  vertical  diaphragm  divides  the  pontoon  longitudinally  into  two 
compartments  each  12  feet  wide  and  50  feet  long.  In  the  lower  part 
of  each  of  these  compartments  there  is  water  ballast,  the  surface  of  the 
water  being  free  to  move. 

Determine  the  position  of  the  centre  of  gravity  of  the  pontoon  that  it 
may  be  stable  for  small  displacements. 

(7)  Define  "metacentric  height"  and  show  how  to  obtain  it  graphically 
or  otherwise.     A  ship  of  16,000  tons  displacement  is  600  feet  long,  60  feet 
beam,  and  26  feet  draught.   A  coefficient  of  ^  may  be  taken  in  the  moment 
of  inertia  term  instead  of  ^  to  allow  for  the  water-line  section  not  being 
a  rectangle.     The  depth  of  the  centre  of  buoyancy  from  the  water  line  is 
10  feet.     Find  the  height  of  the  metacentre  above  the  water  line  and 
determine  the  position  of  the  centre  of  gravity  to  give  a  metacentric  height 
of  18  inches.     [Lond.  Un.  1906.] 

(8)  The  total  weight  of  a  fully  loaded  ship  is  5000  tons,  the  water  line 
encloses  an  area  of  9000  square  feet,  and  the  sides  of  the  ship  are  vertical 
at  the  water  line.     The  ship  was  loaded  in  fresh  water.     Find  the  change 
in  the  depth  of  immersion  after  the  ship  has  been  sufficiently  long  at  sea  to 
burn  500  tons  of  coal. 

Weight  of  1  cubic  foot  of  fresh  water  62£  Ibs. 
Weight  of  1  cubic  foot  of  salt  water  64  Ibs. 


CHAPTER    III. 

FLUIDS  IN  MOTION. 

28.  Steady  motion. 

The  motion  of  a  fluid  is  said  to  be  steady  or  permanent,  when 
the  particles  which  succeed  each  other  at  any  point  whatever 
have  the  same  density  and  velocity,  and  are  subjected  to  the  same 
pressure. 

In  practice  it  is  probably  very  seldom  that  such  a  condition  of 
flow  is  absolutely  realised,  as  even  in  the  case  of  the  water  flowing 
steadily  along  a  pipe  or  channel,  except  at  very  low  velocities,  the 
velocities  of  succeeding  particles  of  water  which  arrive  at  any 
point  in  the  channel,  are,  as  will  be  shown  later,  not  the  same 
either  in  magnitude  or  direction. 

For  practical  purposes,  however,  it  is  convenient  to  assume 
that  if  the  rate  at  which  a  fluid  is  passing  through  any  finite  area 
is  constant,  then  at  all  points  in  the  area  the  motion  is  steady. 

For  example,  if  a  section  of  a  stream  be  taken  at  right  angles 
to  the  direction  of  flow  of  the  stream,  and  the  mean  rate  at  which 
water  flows  through  this  section'  is  constant,  it  is  convenient 
to  assume  that  at  any  point  in  the  section,  the  velocity  always 
remains  constant  both  in  magnitude  and  direction,  although  the 
velocity  at  different  points  may  not  be  the  same. 

Mean  velocity.  The  mean  velocity  through  the  section,  or  the 
mean  velocity  of  the  stream,  is  equal  to  the  quantity  of  flow  per 
unit  time  divided  by  the  area  of  the  section. 

29.  Stream  line  motion. 

The  particles  of  a  fluid  are  generally  regarded  as  flowing  along 
definite  paths,  or,  in  other  words,  the  fluid  may  be  supposed  to 
flow  in  thread-like  filaments,  and  when  the  motion  is  steady  these 
filaments  may  be  supposed  to  be  fixed  in  position. 

In  a  pipe  or  channel  of  constant  section,  the  filaments  are 
generally  supposed  to  be  parallel  to  the  sides  of  the  channel. 


38 


HYDRAULICS 


30.     Definitions  relating  to  flow  of  water. 

Pressure  head.     The  pressure  head  at  a  point  in  a  fluid  at  rest 
has  been  denned  as  the  vertical  distance  of  the  point  from  the  free 

surface  of  the  fluid,  and  is  equal  to  — ,  where  p  is  the  pressure  per 

sq.  foot  and  w  is  weight  per  cubic  foot  of 
the  fluid.  Similarly,  the  pressure  head  at 
any  point  in  a  moving  fluid  at  which  the 

/Y\ 

pressure  is  p  Ibs.  per  sq.  foot,  is  —  feet, 

and  if  a  vertical  tube,  called  a  piezometer 
tube,  Fig.  31,  be  inserted  in  the  fluid,  it 
will  rise  in  the  tube  to  a  height  h,  which 
equals  the  pressure  head  above  the  atmo- 
spheric pressure.  If  p  is  the  pressure  per 
sq.  foot,  above  the  atmospheric  pressure, 

h  =  — ,  but  if  p  is  the  absolute  pressure  per   - 
sq.  foot,  and  pA  the  atmospheric  pressure, 


Datum,],  Level 


Fig.  31. 


w      w 

Velocity  head.  If  through  a  small  area  around  the  point  B, 
the  velocity  of  the  fluid  is  v  feet  per  second,  the  velocity  head  is 

5- ,  g  being  the  acceleration  due  to  gravity  in  feet  per  second  per 

second. 

Position  head.  If  the  point  B  is  at  a  height  z  feet  above  any 
convenient  datum  level,  the  position  head  of  the  fluid  at  B  above 
the  given  datum  is  said  to  be  z  feet. 

31.  Energy  per  pound  of  water  passing  any  section  in 
a  stream  line. 

The  total  amount  of  work  that  can  be  obtained  from  every 
pound  of  water  passing  the  point  B,  Fig.  31,  assuming  it  can  fall  to 
the  datum  level  and  that  no  energy  is  lost,  is 

£  +  I"  +  z  ft.  Ibs. 
w     20 

Proof.  Work  available  due  to  pressure  head.  That  the  work 
which  can  be  done  by  the  pressure  head  per  pound  is  -  foot 

pounds  can  be  shown  as  follows. 

Imagine  a  piston  fitting  into  the  end  of  a  small  tube  of  cross 
sectional  area  a,  in  which  the  pressure  is  h  feet  of  water  as  in 


UNIVERSITY 


FLUIDS    IN    MOTION 


Fig.  32,  and  let  a  small  quantity  <3Q  cubic  feet  of  water  enter  the 
tube  and  move  the  piston  through  a  small  dis- 
tance dx. 

Then  dQ=a.dx. 

The  work  done   on  the  piston  as  it  enters 
will  be 

w  .  h  .  a  .  dx  =  w  . 


1 


But  the  weight  of  dQ  cubic  feet  is  w  .  8Q  pounds,  Fi8-  32- 

and  the  work  done  per  pound  is,  therefore,  h,  or  —  foot  pounds. 

w 

A  pressure  head  h  is  therefore  equivalent  to  h  foot  pounds  of 
energy  per  pound  of  water. 

Work  available  due  to  velocity.  When  a  body  falls  through 
a  height  h  feet,  the  work  done  on  the  body  by  gravity  is  h  foot 
pounds  per  pound.  It  is  shown  in  books  on  mechanics  that  if  the 
body  is  allowed  to  fall  freely,  that  is  without  resistance,  the 
velocity  the  body  acquires  in  feet  per  second  is 

v=  \/2gh, 

v*     z.  * 

Yg  =  h' 

And  ,since  no  resistance  is  offered  to  the  motion,  the  whole  of 
the  work  done  on  the  body  has  been  utilised  in  giving  kinetic 

energy  to  it,  and  therefore  the  kinetic  energy  per  pound  is  ~-  . 

In  the  case  of  the  fluid  moving  with  velocity  v,  an  amount  of 

v2 
energy  equal  to  ~-  foot  pounds  per  pound  is  therefore  available 

before  the  velocity  is  destroyed. 

Work  available  due  to  position.  If  a  weight  of  one  pound 
falls  through  the  height  z  the  work  done  on  it  by  gravity  will  be 
z  foot  pounds,  and,  therefore,  if  the  fluid  is  at  ^height  z  feet  above 
any  datum,  as  for  example,  water  at  a  given  height  above  the 
sea  level,  the  available  energy  on  allowing  the  fluid  to  fall  to 
the  datum  level  is  z  foot  pounds  per  jpund. 

32.     Bernouilli's  theorem. 

In  a  steady  moving  stream  of  an  incompressible  fluid  in  which 
the  particles  of  fluid  are  moving  in  stream  lines,  and  there  is  no 
loss  by  friction  or  other  causes 

p      v2 

+  o~  +  z 
w     2g 

is  constant  for  all  sections  of  the  stream.    This  is  a  most  important 
theorem  and  should  be  carefully  studied  by  the  reader. 


40 


HYDRAULICS 


It  has  been  shown  in  the  last  paragraph  that  this  expression 
represents  the  total  amount  of  energy  per  pound  of  water  flowing 
through  any  section  of  a  stream,  and  since,  between  any  two 
points  in  the  stream  no  energy  is  lost,  by  the  principle  of  the 
conservation  of  energy  it  can  at  once  be  inferred  that  this 
expression  must  be  constant  for  all  sections  of  a  steady  flowing 
stream.  A  more  general  proof  is  as  follows. 

Let  DB,  Fig.  33,  be  the  path  of  a  particle  of  the  fluid. 


Fig.  33. 

Imagine  a  small  tube  to  be  surrounding  DE,  and  let  the  flow 
in  this  be  steady,  and  let  the  sectional  area  of  the  tube  be  so  small 
that  the  velocity  through  any  section  normal  to  DE  is  uniform. 

Then  the  amount  of  fluid  that  flows  in  at  D  through  the  area 
AB  equals  the  amount  that  flows  out  at  E  through  the  area  OF. 

Let  pv  and  17D,  and  pE  and  VE  be  the  pressures  and  velocities  at 
D  and  E  respectively,  and  A  and  a  the  corresponding  areas  of  the 
tube. 

Let  z  be  the  height  of  D  above  some  datum  and  zl  the  height 
of  E. 

Then,  if  a  quantity  of  fluid  ABAiBi  equal  to  3Q  enters  at  D, 
and  a  similar  quantity  CFCiFi  leaves  at  E,  in  a  time  9£,  the 
velocity  at  D  is 


and  the  velocity  at  E  is 


Aut 


—  . 
aot 


The  kinetic  energy  of  the  quantity  of  fluid  uQ,  entering  at  D 


FLUIDS   IN  MOTION  41 

and  that  of  the  liquid  leaving  at  E 


Since  the  flow  in  the  tube  is  steady,  the  kinetic  energy  o%the 
portion  ABCF  does  not  alter,  and  therefore  the  increase  of  the 
kinetic  energy  of  the  quantity  dQ 

w  .c 


The  work  done  by  gravity  is  the  same  as  if  ABBiAi  fell  to 
i  and  therefore  equals 


The  total  pressure  on  the  area  AB  is  pD  .  A,  and  the  work  done 
at  D  in  time  dt 


and  the  work  done  by  the  pressure  at  E  in  time  t 


But  the  gain  of  kinetic  energy  must  equal  the  work  done,  and 
therefore 


.  0E2  -  V)  -  wdQ,  (z  -  Zi)  +  pD  9Q  -  pE  dQ. 

From  which 

V_V_  ,PD_^E 

2g      2g         ~Zl+  w      w1 

VE       PE  ^D2      PD 

or  ^  +  r*+  Zl  =  7?-+^  +  z  =  constant. 

Zg      w  2g      w 

From  this  theorem  it  is  seen  that,  if  at  points  in  a  steady 
moving  stream,  a  vertical  ordinate  equal  to  the  velocity  head  plus 
the  pressure  head  is  erected,  the  upper  extremities  of  these 
ordinates  will  be  in  the  same  horizontal  plane,  at  a  height  H 

T)          V^ 

equal  to  —  +  ~-  +  z  above  the  datum  level. 
w     2g 

Mr  Froude*  has  given  some  very  beautiful  experimental  illus- 
trations of  this  theorem. 

In  Fig.  34  water  is  taken  from  a  tank  or  reservoir  in  which 
the  water  is  maintained  at  a  constant  level  by  an  inflowing 
stream,  through  a  pipe  of  variable  diameter  fitted  with  tubes 
at  various  points.  Since  the  pipe  is  short  it  may  be  supposed  to 
be  frictionless.  If  the  end  of  the  pipe  is  closed  the  water  will  rise 
in  all  the  tubes  to  the  same  level  as  the  water  in  the  reservoir,  but 
if  the  end  C  is  opened,  water  will  flow  through  the  pipe  and  the 
water  surfaces  in  the  tubes  will  be  found  to  be  at  different  levels. 
*  British  Assoc.  Report  1875. 


42 


HYDRAULICS 


The  quantity  of  water  flowing  per  second  through  the  pipe  can  be 
measured,  and  the  velocities  at  A,  B,  and  C  can  be  found  by 
dividing  this  quantity  by  the  cross-sectional  areas  of  the  pipe  at 
these  points. 


Fig.  34. 

If  to  the'  head  of  water  in  the  tubes  at  A  and  B  the  ordinates 
£-  and  —-  be  added  respectively,  the  upper  extremities  of  these 

ordinates  will  be  practically  on  the  same  level  and  nearly  level 
with  the  surface  of  the  water  in  the  reservoir,  the  small  difference 
being  due  to  frictional  and  other  losses  of  energy. 

At  C  the  pressure  is  equal  to  the  atmospheric  pressure,  and 
neglecting  friction  in  the  pipe,  the  whole  of  the  work  done  by 
gravity  on  any  water  leaving  the  pipe  while  it  falls  from  the 
surface  of  the  water  in  the  reservoir  through  the  height  H,  which 
is  H  ft.  Ibs.  per  pound,  is  utilised  in  giving  velocity  of  motion  to 
the  water,  and,  as  will  be  seen  later,  in  setting  up  internal  motions. 

Neglecting  these  resistances, 


Due  to  the  neglected  losses,  the  actual  velocity  measured  will  be 
less  than  vc  as  calculated  from  this  equation. 

If  at  any  point  D  in  the  pipe,  the  sectional  area  is  less  than  the 
area  at  C,  the  velocity  will  be  greater  than  vc,  and  the  pressure 
will  be  less  than  the  atmospheric  pressure. 

If  v  is  the  velocity  at  any  section  of  the  pipe,  which  is  supposed 
to  be  horizontal,  the  absolute  pressure  head  at  that  section  is 

£.  =  ^+H-—  =  2^  +  ^-  — 
w      w  2g     w      2g     2g' 

pa  being  the  atmospheric  pressure  at  the  surface  of  the  water  in 
the  reservoir. 

At  D  the  velocity  V-Q  is  greater  than  vc  and  therefore  p&  is  less 


FLUIDS    IN   MOTION 


43 


than  pa.     If  coloured  water  be  put  into  the  vessel  E,  it  will  rise  in 
the  tube  DE  to  a  height 


w 


w 


2<7      2<7* 

If  the  area  at  the  section  is  so  small,  that  p  becomes  negative,  the 
fluid  will  be  in  tension,  and  discontinuity  of  flow  will  take  place. 

If  the  fluid  is  water  which  has  been  exposed  to  the  atmosphere 
and  which  consequently  contains  gases  in  solution,  these  gases 
will  escape  from  the  water  if  the  pressure  becomes  less  than  the 
tension  of  the  dissolved  gases,  and  there  will  be  discontinuity  even 
before  the  pressure  becomes  zero. 

Figs.  35  and  36  show  two  of  Froude's  illustrations  of  the 
theorem. 


Fig.  35. 


Fig.  36. 


At  the  section  B,  Fig.  36,  the  pressure  head  is  hE  and  the 
velocity  head  is 


If  a  is  the  section  of  the  pipe  at  A,  and  a\  at  B,  since  there 
is  continuity  of  flow, 

2  2 

and  2^  +  JiA  =  ^-  +  hE  =  h. 

If  now  a  is  made  so  that 


the  pressure  head  hA  becomes  equal  to  the  atmospheric  pressure, 
and  the  pipe  can  be  divided  at  A,  as  shown  in  the  figure. 

Professor  Osborne  Reynolds  devised  an  interesting  experiment, 
to  show  that  when  the  velocity  is  high,  the  pressure  is  small. 

He  allowed  water  to  flow  through  a  tube  f  inch  diameter 
under  a  high  pressure,  the  tube  being  diminished  at  one  section  to 
0'05  inch  diameter. 


44  HYDRAULICS 

At  this  diminished  section,  the  velocity  was  very  high  and  the 
pressure  fell  so  low  that  the  water  boiled  and  made  a  hissing 
noise. 

33.     Venturi  meter. 

An  application  of  Bernoulli's  theorem  is  found  in  the  Yenturi 
meter,  as  invented  by  Mr  Clemens  Herschel*.  The  meter  takes 
its  name  from  an  Italian  philosopher  who  in  the  last  decade  of  the 
l£th  century  made  experiments  upon  the  flow  of  water  through 
conical  pipes.  In  its  usual  form  the  Yenturi  meter  consists  of  two 
truncated  conical  pipes  connected  together  by  a  short  cylindrical 
pipe  called  the  throat,  as  shown  in  Figs.  37  and  38.  The  meter  is 
inserted  horizontally  in  a  line  of  piping,  the  diameter  of  the  large 
ends  of  the  frustra  being  equal  to  that  of  the  pipe. 

Piezometer  tubes  or  other  pressure  gauges  are  connected  to 
the  throat  and  to  one  or  both  of  the  large  ends  of  the  cones. 

Let  a  be  the  area  of  the  throat. 

Let  di  be  the  area  of  the  pipe  or  the  large  end  of  the  cone 
at  A. 

Let  a2  be  the  area  of  the  pipe  or  the  large  end  of  the  cone 
atC. 

Let  p  be  the  pressure  head  at  the  throat. 

Let  pi  be  the  pressure  head  at  the  up-stream  gauge  A. 

Let  £>2  be  the  pressure  head  at  the  down-stream  gauge  C. 

Let  H  and  HI  be  the  differences  of  pressure  head  at  the  throat 
and  large  ends  A  and  C  of  the  cone  respectively,  or 

H  =  2-'-£, 

w      w 

and  H,  =  »-£. 

W          W 

Let  Q  be  the  flow  through  the  meter  in  cubic  feet  per  sec. 
Let  v  be  the  velocity  through  the  throat. 
Let  Vi  be  the  velocity  at  the  up-stream  large  end  of  cone  A. 
Let  vz  be  the  velocity  at  the  down-stream  large  end  of  cone  C. 
Then,  assuming  Bernouilli's  theorem,  and  neglecting  friction, 


and 


.  =  = 

w     2g     w     2g     w 

TT    v9  —  v^ 


If  v2  is  equal  to  Vi,  p2  is  theoretically  equal  to  pi,  but  there  is 
always  in  practice  a  slight  loss  of  head  in  the  meter,  the  difference 
Pi  -  p*  being  equal  to  this  loss  of  head. 

*  Transactions  Am.S.C.E.,  1887. 


FLUIDS   IN   MOTION 


45 


'     1 


46 


HYDRAULICS 


The  velocity  v  is  —  ,  and  Vi  is  —  . 
a,  a\ 

Therefore  Q2  (~2  ~  ^)  =  20 .  H, 

\tl         QJ\  / 


and 


Due  to  friction,  and  eddy  motions  that  may  be  set  up  in  the 
meter,  the  discharge  is  slightly  less  than  this  theoretical  value,  or 


Jc  being  a  coefficient  which  has  to  be  determined  by  experiment. 

For  a  meter  having  a  diameter  of  25'5  inches  at  the  throat  and 
54  inches  at  the  large  end  of  the  cone,  Herschel  found  the 
following  values  for  &,  given  in  Table  III,  so  that  the  coefficient 
varies  but  little  for  a  large  variation  of  H. 

TABLE  III. 


Herschel 

Coker 

Hfeet 

k 

Discharge 

k 

in  cu.  ft. 

1 

•995 

•0418 

•9494 

2 

•992 

•0319 

•9587 

6 

•985 

•0254 

•9572 

12 

•9785 

•0185 

•9920 

18 

•977 

•0096 

1-2021 

23 

•970 

•0084 

1-3583 

Professor  Coker*,  from  careful  experiments  on  an  exceedingly 
well  designed  small  Yenturi  meter,  Fig.  38,  the  area  of  the  throat 
of  which  was  '014411  sq.  feet,  found  that  for  small  flows  the 
coefficient  was  very  variable  as  shown  in  Table  III. 

These  results  show,  as  pointed  out  by  Professor  Coker  from  an 
analysis  of  his  own  and  Herschel's  experiments  on  meters  of 
various  sizes,  that  in  large  Venturi  meters,  the  discharge  is  very 
approximately  proportional  to  the  square  root  of  the  head,  but  for 
small  meters  it  only  follows  this  law  for  high  heads,  and  for  low 
heads  they  require  special  calibration. 

Example.  A  Venturi  meter  having  a  diameter  at  the  throat  of  36  inches  is 
inserted  in  a  9  foot  diameter  pipe. 

The  pressure  head  at  the  throat  gauge  is  20  feet  of  water  and  at  the  pipe  gauge 
is  26  feet. 

*  Canadian  Society  of  Civil  Engineers,  1902. 


FLUIDS   IN   MOTION 

Find  the  discharge,  and  the  velocity  of  flow  through  the  throat. 
The  area  of  the  pipe  is  63-5  sq.  feet. 

throat      7-05 
The  difference  in  pressure  head  at  the  two  gauges  is  6  feet. 

63-5x7-05 
Therefore  Q  = 


47 


=  137  c.  ft.  per  second. 
The  velocity  of  flow  in  the  pipe  is  2-15  ft.  per  sec. 

,,  ,,  through  the  throat  is  19-4  ft.  per  sec. 

34.     Steering  of  canal  boats. 

An  interesting  application  of  Bernoulli's  theorem  is  to  show 
the  effect  of  speed  and  position  on  the  steering  of  a  canal  boat. 

When  a  boat  is  moved  at  a  high  velocity  along  a  narrow 
and  shallow  canal,  the  boat  tends  to  leave  behind  it  a  hollow 
which  is  filled  by  the  water  rushing  past  the  boat  as  shown 
in  Figs.  39  and  40,  while  immediately  in  front  of  the  boat  the 
impact  of  the  bow  on  the  still  water  causes  an  increase  in  the 
pressure  and  the  water  is  "  piled  up  "  or  is  at  a  higher  level  than 
the  still  water,  and  what  is  called  a  bow  wave  is  formed. 

Fig.  39.  Fig.  41. 


A 

Fig.  40. 

Let  it  be  assumed  that  the  water  moves  past  the  boat  in 
stream  lines. 

If  vertical  sections  are  taken  at  E  and  F,  and  the  points  E  and 
F  are  on  the  same  horizontal  line,  by  Bernoulli's  theorem 

w      2g      w      2g  ' 

At  E  the  water  is  practically  at  rest,  and  therefore  VE  is 
zero,  and 

w  ~  w      2g' 
The  surface  at  E  will  therefore  be  higher  than  at  F. 


48  HYDRAULICS 

When  the  boat  is  at  the  centre  of  the  canal  the  stream  lines  on 
both  sides  of  the  boat  will  have  the  same  velocity,  but  if  the  boat 
is  nearer  to  one  bank  than  the  other,  as  shown  in  the  figures,  the 
velocity  v?>  of  the  stream  lines  between  the  boat  and  the  nearer 
bank,  Fig.  41,  will  be  higher  than  the  velocity  VF  on  the  other 
side.  But  for  each  side  of  the  boat 

PV=PV  +  ^  =  PP'  +  ^F2 

w      w      2g      w       2g  ' 

And  since  v$>  is  greater  than  VF,  the  pressure  head  pv  is 
greater  than  pF',  or  in  other  words  the  surface  of  the  water  at 
the  right  side  D  of  the  boat  will  be  higher  than  on  the  left  side  B. 

The  greater  pressure  011  the  right  side  D  tends  to  push  the 
boat  towards  the  left  bank  A,  and  at  high  speeds  considerably 
increases  the  difficulty  of  steering. 

This  difficulty  is  diminished  if  the  canal  is  made  sufficiently 
deep,  so  that  flow  can  readily  take  place  underneath  the  boat. 

35.    Extension  of  Bernouilli's  theorem. 

In  deducing  this  theorem  it  has  been  assumed  that  the  fluid 
is  a  perfect  fluid  moving  with  steady  motion  and  that  there  are  no 
losses  of  energy,  by  friction  of  the  surfaces  with  which  the  fluid 
may  be  in  contact,  or  by  the  relative  motion  of  consecutive  ele- 
ments of  the  fluid,  or  due  to  internal  motions  of  the  fluid. 

In  actual  cases  the  value  of 

73      v2 

~  +  ^r  +  z 
w     2g 

diminishes  as  the  motion  proceeds. 

If  hf  is  the  loss  of  head,  or  loss  of  energy  per  pound  of  fluid, 
between  any  two  given  points  A  and  B  in  the  stream,  then  more 
generally 


ZA  =  ls          f    ...............          . 

w      2g  w      2g 

EXAMPLES. 

(1)  The  diameter  of  the  throat  of  a  Venturi  meter  is  f  inch,  and  of 
the  pipe  to  which  it  is  connected  If  inches.     The  discharge  through  the 
meter  in  20  minutes  was  found  to  be  314  gallons. 

The    difference    in    pressure    head  at  the  two  gauges  was  49  feet. 
Determine  the  coefficient  of  discharge. 

(2)  A  Venturi  meter  has  a  diameter  of  4  ft.  in  the  large  part  and 
1-25  ft.  in  the  throat.     With  water  flowing  through  it,  the  pressure  head  is 
100  ft.  in  the  large  part  and  97  ft.  at  the  throat.     Find  the  velocity  in  the 
small  part  and  the  discharge  through  the  meter.     Coefficient  of  meter 
taken  as  unity. 


FLUIDS   IN   MOTION  49 

(3)  A  pipe  AB,  100  ft.  long,  has  an  inclination  of  1  in  5.    The  head  due 
to  the  pressure  at  A  is  45  ft.,  the  velocity  is  3  ft.  per  second,  and  the  section 
of  the  pipe  is  3  sq.  ft.     Find  the  head  due  to  the  pressure  at  B,  where  the 
section  is  1|  sq.  ft.     Take  A  as  the  lower  end  of  the  pipe. 

(4)  The  suction  pipe  of  a  pump  is  laid  at  an  inclination  of  1  hi  5,  and 
water  is  pumped  through  it  at  6  ft.  per  second.     Suppose  the  air  hi  the 
water  is  disengaged  if  the  pressure  falls  to  more  than   10  Ibs.  below 
atmospheric  pressure.     Then  deduce  the  greatest  practicable  length  of 
suction  pipe.     Friction  neglected. 

(5)  Water  is  delivered  to  an  inward-flow  turbine  under  a  head  of  100  feet 
(see   Chapter   IX).     The  pressure  just  outside  the  wheel  is  25  Ibs.  per 
sq.  inch  by  gauge. 

Find  the  velocity  with  which  the  water  approaches  the  wheel.  Friction 
neglected. 

(6)  A  short  conical  pipe  varying  in  diameter  from  4'  6"  at  the  large  end 
to  2  feet  at  the  small  end  forms  part  of  a  horizontal  water  main.     The 
pressure  head  at  the  large  end  is  found  to  be  100  feet,  and  at  the  small  end 
96-5  feet. 

Find  the  discharge  through  the  pipe.     Coefficient  of  discharge  unity. 

(7)  Three  cubic  feet  of  water  per  second  flow  along  a  pipe  which  as  it 
falls  varies  in  diameter  from  6  inches  to  12  inches.     In  50  feet  the  pipe 
falls  12  feet.     Due  to  various  causes  there  is  a  loss  of  head  of  4  feet. 

Find  (a)  the  loss  of  energy  in  foot  pounds  per  minute,  and  in  horse- 
power, and  the  difference  in  pressure  head  at  the  two  points  50  feet  apart.. 
(Use  equation  1,  section  35.) 

(8)  A  horizontal  pipe  in  which  the  sections  vary  gradually  has  sections 
of  10  square  feet,  1  square  foot,  and  10  square  feet  at  sections  A,  B,  and  C. 
The  pressure  head  at  A  is  100  feet,  and  the  velocity  3  feet  per  second. 
Find  the  pressure  head  and  velocity  at  B. 

Given  that  in  another  case  the  difference  of  the  pressure  heads  at  A 
and  B  is  2  feet.     Find  the  velocity  at  A. 

(9)  A  Venturi  meter  in  a  water  main  consists  of  a  pipe  converging  to 
the  throat  and  enlarging  again  gradually.     The  section  of  main  is  9  sq.  ft. 
and  the  area  of  throat  1  sq.  ft.     The  difference  of  pressure  in  the  main  and 
at  the  throat  is  12  feet  of  water.     Find  the  discharge  of  the  main  per  hour. 

(10)  If  the  inlet  area  of  a  Venturi  meter  is  n  times  the  throat  area,  and 
v  and  p  are  the  velocity  and  pressure  at  the  throat,  and  the  inlet  pressure 
is  mp,  show  that — 


and  show  that  if  p  and  mp  are  observed,  v  can  be  found. 


L.  H. 


CHAPTER   IV. 

FLOW  OF  WATER  THROUGH   ORIFICES  AND 
OVER  WEIRS. 

36.     Plow  of  fluids  through  orifices. 

The  general  theory  of  the  discharge  of  fluids  through  orifices, 
as  for  example  the  flow  of  steam  and  air,  presents  considerable 
difficulties,  and  is  somewhat  outside  the  scope  of  this  treatise. 
Attention  is,  therefore,  confined  to  the  problem  of  determining  the 
quantity  of  water  which  flows  through  a  given  orifice  in  a  given 
time,  and  some  of  the  phenomena  connected  therewith. 

In  what  follows,  it  is  assumed  that  the  density  of  the  fluid  is 
constant,  the  effect  of  small  changes  of  temperature  and  pressure 
in  altering  the  density  being  thus  neglected. 

Consider  a  vessel,  Fig.  42,  filled  with  water,  the  free  surface  of 
which  is  maintained  at  a  constant  level ;  in  the  lower  part  of  the 
vessel  there  is  an  orifice  AB. 


Fig.  42. 


Let  it  be  assumed  that  although  water  flows  into  the  vessel  so 
as  to  maintain  a  constant  head,  the  vessel  is  so  large  that  at  some 
surface  CD,  the  velocity  of  flow  is  zero. 

Imagine  the  water  in  the  vessel  to  be  divided  into  a  number  of 
stream  lines,  and  consider  any  stream  line  EF. 

Let  the  velocities  at  E  and  F  be  VE  and  VF,  the  pressure  heads 
hE  and  hF,  and  the  position  heads  above  some  datum,  ZE  and  ZF, 
respectively. 


FLOW  THROUGH  ORIFICES  51 

Then,  applying  Bernoulli's  theorem  to  the  stream  line  EF, 

If  v?  is  zero,  then 

2 

But  from  the  figure  it  is  seen  that 
is  equal  to  h,  and  therefore 


or  VE 

Since  HE  is  the  pressure  head  at  E,  the  water  would  rise  in 
a  tube  having  its  end  open  at  E,  a  height  7iE,  and  h  may  thus 
be  called — following  Thomson — the  fall  of  "free  level  for  the 
point  E." 

At  some  section  GK  near  to  the  orifice  the  stream  lines  are  all 
practically  normal  to  the  section,  and  the  pressure  head  will  be 
equal  to  the  atmospheric  pressure ;  and  if  the  orifice  is  small  the  fall 
of  free  level  for  all  the  stream  lines  is  H,  the  distance  of  the  centre 
of  the  section  GK  below  the  free  surface  of  the  water.  If  the 
orifice  is  circular  and  sharp-edged,  as  in  Figs.  44  and  45,  the  section 
GK  is  at  a  distance,  from  the  plane  of  the  orifice,  about  equal  to 
its  radius.  For  vertical  orifices,  and  small  horizontal  orifices, 
H  may  be  taken  as  equal  to  the  distance  of  the  centre  of  the 
orifice  below  the  free  surface. 

The  theoretical  velocity  of  flow  through  the  small  section  GK 
is,  therefore,  the  same  for  all  the  stream  lines,  and  equal  to  the 
velocity  which  a  body  will  acquire,  in  falling,  in  a  vacuum, 
through  a  height,  equal  to  the  depth  of  the  centre  of  the  orifice 
below  the  free  surface  of  the  water  in  the  vessel. 

The  above  is  Thomson's  proof  of  Torricelli's  theorem,  which 
was  discovered  experimentally,  by  him,  about 
the  middle  of  the  17th  century. 

The  theorem  is  proved  experimentally  as 
follows. 

If  the  aperture  is  turned  upwards,  as  in 
Fig.  43,  it  is  found  that  the  water  rises 
nearly  to  the  level  of  the  water  in  the  vessel, 
and  it  is  inferred,  that  if  the  resistance  of  the 
air  and  of  the  orifice  could  be  eliminated,  the 
jet  would  rise  exactly  to  the  level  of  the 
surface  of  the  water  in  the  vessel. 


52 


HYDRAULICS 


Other  experiments  described  on  pages  54—56,  also  show  that, 
with  carefully  constructed  orifices,  the  mean  velocity  through  the 
orifice  differs  from  \/2^H  by.  a  very  small  quantity. 

37.     Coefficient  of  contraction  for  sharp-edged  orifice. 

If  an  orifice  is  cut  in  the  flat  side,  or  in  the  bottom  of  a  vessel, 
and  has  a  sharp  edge,  as  shown  in  Figs.  44  and  45,  the  stream  lines 
set  up  in  the  water  approach  the  orifice  in  all  directions,  as  shown 
in  the  figure,  and  the  directions  of  flow  of  the  particles  of  water, 
except  very  near  the  centre,  are  not  normal  to  the  plane  of  the 
orifice,  but  they  converge,  producing  a  contraction  of  the  jet. 


i 


Fig.  44.  Fig.  45. 

At  a  small  distance  from  the  orifice  the  stream  lines  become 
practically  parallel,  but  the  cross  sectional  area  of  the  jet  is 
considerably  less  than  the  area  of  the  orifice. 

If  w  is  the  area  of  the  jet  at  this  section  and  a  the  area  of  the 

orifice  the  ratio  -  is  called  the  coefficient  of  contraction  and  may 
a 

be  denoted  by  c.  Weisbach  states,  that  for  a  circular  orifice,  the 
jet  has  a  minimum  area  at  a  distance  from  the  orifice  slightly  less 
than  the  radius  of  the  orifice,  and  defines  the  coefficient  of 
contraction  as  this  area  divided  by  the  area  of  the  orifice.  For  a 
circular  orifice  he  gives  to  c  the  value  0*64.  Recent  careful 
measurements  of  the  sections  of  jets  from  horizontal  and  vertical 
sharp-edged  circular  and  rectangular  orifices,  by  Bazin,  the 
results  of  some  of  which  are  shown  in  Table  IY,  show,  however, 
that  the  section  of  the  jet  diminishes  continuously  and  in  fact  has 
no  minimum  value.  Whether  a  minimum  occurs  for  square  orifices 
is  doubtful. 

The  diminution  in  section  for  a  greater  distance  than  that 
given  by  Weisbach  is  to  be  expected,  for,  as  the  jet  moves  away 
from  the  orifice  the  centre  of  the  jet  falls,  and  the  theoretical 
velocity  becomes  */2g  (H  +  y),  y  being  the  vertical  distance  between 
the  centre  of  the  orifice  and  the  centre  of  the  jet. 


FLOW   THROUGH   ORIFICES  53 

At  a  small  distance  away  from  the  orifice,  however,  the  stream 
lines  are  practically  parallel,  and  very  little  error  is  introduced  in 
the  coefficient  of  contraction  by  measuring  the  stream  near  the 
orifice. 

Poncelet  and  Lesbros  in  1828  found,  for  an  orifice  "20  m.  square, 
a  minimum  section  of  the  jet  at  a  distance  of  '3  m.  from  the  orifice 
and  at  this  section  c  was  '563.  M.  Bazin,  in  discussing  these 
results,  remarks  that  at  distances  greater  than  0*3  m.  the  section 
becomes  very  difficult  to  measure,  and  although  the  vein  appears 
to  expand,  the  sides  become  hollow,  and  it  is  uncertain  whether 
the  area  is  really  diminished. 

Complete  contraction.  The  maximum  contraction  of  the  jet 
takes  place  when  the  orifice  is  sharp  edged  and  is  well  removed 
from  the  sides  and  bottom  of  the  vessel.  In  this  case  the  contrac- 
tion is  said  to  be  complete.  Experiments  show,  that  for  complete 
contraction  the  distance  from  the  orifice  to  the  sides  or  bottom  of 
the  vessel  should  not  be  less  than  one  and  a  half  to  twice  the  least 
diameter  of  the  orifice. 

Incomplete  or  suppressed  contraction.  An  example  of  incom- 
plete contraction  is  shown  in  Fig.  46,  the  lower  edge  of  the 
rectangular  orifice  being  made  level  with  the  bottom  of  the  vessel. 
The  same  effect  is  produced  by  placing  a  horizontal  plate  in 
the  vessel  level  with  the  bottom  of  the  orifice.  The  stream 
lines  at  the  lower  part  of  the  orifice  are  normal  to  its  plane 
and  the  contraction  at  the  lower  edge  is  consequently  suppressed. 


Fig.  46.  Fig.  47. 

Similarly,  if  the  width  of  a  rectangular  orifice  is  made  equal 
to  that  of  the  vessel,  or  the  orifice  abed  is  provided  with  side  walls 
as  in  Fig.  47,  the  side  or  lateral  contraction  is  suppressed.  In  any 
case  of  suppressed  contraction  the  discharge  is  increased,  but,  as 
will  be  seen  later,  the  discharge  coefficient  may  vary  more  than 
when  the  contraction  is  complete.  To  suppress  the  contraction 
completely,  the  orifice  must  be  made  of  such  a  form  that  the 
stream  lines  become  parallel  at  the  orifice  and  normal  to  its  plane. 


54 


HYDRAULICS 


Experimental  determination  of  c.  The  section  of  the  stream 
from  a  circular  orifice  can  be  obtained  with  considerable  accu- 
racy by  the  apparatus  shown  in  Fig.  49,  which  consists  of  a 
ring  having  four  radial  set 
screws  of  fine  pitch.  The 
screws  are  adjusted  until  the 
points  thereof  touch  the  jet. 
M.  Bazin  has  recently  used  an 
octagonal  frame  with  twenty- 
four  set  screws,  all  radiating 
to  a  common  centre,  to  deter- 
mine the  form  of  the  section 
of  jets  from  various  kinds  of 
orifices.  Fig.  48>  Fig.  49. 

The  screws  were  adjusted 

until  they  just  touched  the  jet.  The  frame  was  then  placed  upon 
a  sheet  of  paper  and  the  positions  of  the  ends  of  the  screws 
marked  upon  the  paper.  The  forms  of  the  sections  could  then 
be  obtained,  and  the  areas  measured  with  considerable  accuracy. 
Some  of  the  results  obtained  are  shown  in  Table  IV  and  also  in 
the  section  on  the  form  of  the  liquid  vein. 

38.     Coefficient  of  velocity  for  sharp-edged  orifice. 

The  theoretical  velocity  through  the  contracted  section  is,  as 
shown  in  section  36,  equal  to  \/20H,  but  the  actual  velocity 
vl  is  slightly  less  than  this  due  to  friction  at  the  orifice.  The 

ratio  -  l  =  Jc  is  called  the  coefficient  of  velocity. 

Experimental  determination  of  k.  There  are  two  methods 
adopted  for  determining  k  experimentally. 

First  method.  The  velocity  is  determined  by  measuring  the 
discharge  in  a  given  time  under  a  given  head,  and  the  cross 
sectional  area  w  of  the  jet,  as  explained  in  the  last  paragraph,  is 
also  obtained.  Then,  if  Vi  is  the  actual  velocity,  and  Q  the 
discharge  per  second, 


and 


» 


Second  method.  An  orifice,  Fig.  50,  is  formed  in  the  side  of  a 
vessel  and  water  allowed  to  flow  from  it.  The  water  after  leaving 
the  orifice  flows  in  a  parabolic  curve.  Above  the  orifice  is  fixed 
a  horizontal  scale  on  which  is  a  slider  carrying  a  vertical  scale, 
to  the  bottom  of  which  is  clamped  a  bent  piece  of  wire,  with  a  sharp 


FLOW   THROUGH    ORIFICES 


55 


point.  The  vertical  scale  can  be  adjusted  so  that  the  point  touches 
the  upper  or  lower  surface  of  the  jet,  and  the  horizontal  and  vertical 
distances  of  any  point  in  the  axis  of  the  jet  from  the  centre  of  the 
orifice  can  thus  be  obtained. 


Fig.  50. 

Assume  the  orifice  is  vertical,  and  let  Vi  be  the  horizontal 
velocity  of  flow.  At  a  time  t  seconds  after  a  particle  has  passed 
the  orifice,  the  distance  it  has  moved  horizontally  is 

X  =  Vit (1). 

The  vertical  distance  is 

y  =  \qt\.  (2). 


Therefore 


and 


The  theoretical  velocity  of  flow  is 


Therefore 


It  is  better  to  take  two  values  of  x  and  y  so  as  to  make 
allowance  for  the  plane  of  the  orifice  not  being  exactly  perpen- 
dicular. 

If  the  orifice  has  its  plane  inclined  at  an  angle  6  to  the 
vertical,  the  horizontal  component  of  the  velocity  is  vl  cos  0  and 
the  vertical  component  v\  sin  0. 

At  a  time  t  seconds  after  a  particle  has  passed  the  orifice,  the 
horizontal  movement  from  the  orifice  is, 


and  the  vertical  movement  is, 

y  =  vlsmet  + 
After  a  time  h  seconds        x1  =  v1cosOtlL 


(2). 


56  HYDRAULICS 

Substituting  the  value  of   t  from  (1)  in  (2)  and  ti  from  (3) 
in  (4), 


and,  ^]tan,  +  _  .....................  (6). 

From  (5),  W  =  jt**fO 

g       y-x  tan  0 

Substituting  for  Vi  in  (6), 

=^-^'3.  ........................  (8). 


Having  calculated  tan  0,  sec  0  can  be  found  from  mathematical 
tables,  and  from  (7)  v\  can  be  calculated.     Then 

ft- 


39.     Bazin's  experiments  on  a  sharp-edged  orifice. 

In  Table  IV  are  given  values  of  k  as  obtained  by  Bazin  from 
experiments  on  vertical  and  horizontal  sharp-edged  orifices,  for 
various  values  of  the  head. 

The  section  of  the  jet  at  various  distances  from  the  orifice  was 
carefully  measured  by  the  apparatus  described  above,  and  the 
actual  discharge  per  second  was  determined  by  noting  the  time 
taken  to  fill  a  vessel  of  known  capacity. 

The  mean  velocity  through  any  section  was  then 


Q  being  the  discharge  per  second  and  A  the  area  of  the  section. 

The  fall  of  free  level  for  the  various  sections  was  different,  and 
allowance  is  made  for  this  in  calculating  the  coefficient  k  in  the 
fourth  column. 

Let  y  be  the  vertical  distance  of  the  centre  of  any  section 
below  the  centre  of  the  orifice  ;  then  the  fall  of  free  level  for  that 
section  is  H  +  y  and  the  theoretical  velocity  is 

V2^(HT7). 

The  coefficients  given  in  column  3  were  determined  by  dividing 
the  actual  mean  velocity  through  different  sections  of  the  jet  by 
v  2gH,  the  theoretical  velocity  at  the  centre  of  the  orifice. 

Those  in  column  4  were  found  by  dividing  the  actual  mean 
velocity  through  the  section  by  \/2#  (H  +  y),  the  theoretical 
velocity  at  any  section  of  the  jet. 

The  coefficient  of  column  3  increases  as  the  section  is  taken 
further  from  the  jet,  and  in  nearly  all  cases  is  greater  than  unity. 


FLOW  THROUGH   ORIFICES 


TABLE  IV. 

Sharp-edged  Orifices  Contraction  Complete. 

Table  showing  the  ratio  of  the  area  of  the  jet  to  the  area  of 
the  orifice  at  definite  distances  from  the  orifice,  and  the  ratio  of 
the  mean  velocity  in  the  section  to  \/2#H  and  to  \l2g .  (H  +y), 
H  being  the  head  at  the  centre  of  the  orifice  and  y  the  vertical 
distance  of  the  centre  of  the  section  of  the  jet  from  the  centre  of 
the  orifice. 

Vertical  circular  orifice  0'20  m.  ('656  feet)  diameter,  H  =  '990  m. 
(3-248  feet). 

Coefficient  of  discharge  m,  by  actual  measurement  of  the  flow  is 

m  =  '5977*. 

Distance  of  the  section  Area  of  Jet  Mean  Velocity         Mean  Velocity 

from  the  plane  of  the 


orifice  in  metres 

0-08 

0-13 

0-17 

0-235 

0-335 

0-515 


Area  of  Orince 

=  c 

•6079 
•5971 
•5951 
•5904 
•5830 
•5690 


0-983 
1-001 
1-004 
1-012 
1-025 
1-050 


•998 

•999 

1-003 

1-007 

1-010 


Horizontal    circular    orifice    0'20   m.     ('656    feet)    diameter, 
=  '975m.  (3-198  feet). 

m  =  0'6035. 


0-075 
0-093 
0-110 
0-128 
0-145 
0-163 


0-6003 
0-5939 
0-5824 
0-5734 
0-5658 
0-5597 


1-005 
1-016 
1-036 
1-053 
1-067 
1-078 


0-968 
0-971 
0-982 
0-990 
0-996 
0-998 


Vertical  orifice  '20  m.  ('656  feet)  square,  H  =  '953  m.  (3126  feet). 
m  =  0-6066. 


0-151 
0-175 
0-210 
0-248 
0-302 
0-350 


0-6052 
0-6029 
0-5970 
0-5930 
0-5798 
0-5783 


1-002 
1-006 
1-016 
1-023 
1-046 
1-049 


•997 
1-000 
1-007 
1-010 
1-027 
1-024 


The  real  value  of  the  coefficient  for  the  various  sections  is 
however  that  given  in  column  4. 

For  the  horizontal  orifice,  for  every  section,  it  is  less  than 
unity,  but  for  the  vertical  orifice  it  is  greater  than  unity. 

Bazin's  results  confirm  those  of  Lesbros  and  Poncelet,  who  m 


*  See  section  42. 


58  HYDRAULICS 

1828  found  that  the  actual  velocity  through  the  contracted  section 
of  the  jet,  even  when  account  was  taken  of  the  centre  of  the 
section  of  the  jet  being  below  the  centre  of  the  orifice,  was 
-gV  greater  than  the  theoretical  value. 

This  result  appears  at  first  to  contradict  the  principle  of  the 
conservation  of  energy,  and  Bernoulli's  theorem. 

It  should  however  be  noted  that  the  vertical  dimensions  of  the 
orifice  are  not  small  compared  with  the  head,  and  the  explanation 
of  the  apparent  anomaly  is  no  doubt  principally  to  be  found  in  the 
fact  that  the  initial  velocities  in  the  different  horizontal  filaments 
of  the  jet  are  different. 

Theoretically  the  velocity  in  the  lower  part  of  the  jet  is  greater 
than  \/2#  (H  +  y\  and  in  the  upper  part  less  than  v2<7  (H  +  y). 

Suppose  for  instance  a  section  of  a  jet,  the  centre  of  which  is 
1  metre  below  the  free  surface,  and  assume  that  all  the  filaments 
have  a  velocity  corresponding  to  the  depth  below  the  free  surface, 
and  normal  to  the  section.  This  is  equivalent  to  assuming  that 
the  pressure  in  the  section  of  the  jet  is  constant,  which  is  probably 
not  true. 

Let  the  jet  be  issuing  from  a  square  orifice  of  '2  m.  ('656  feet) 
side,  and  assume  the  coefficient  of  contraction  is  '6,  and  for 
simplicity  that  the  section  of  the  jet  is  square. 

Then  the  side  of  the  jet  is  '1549  metres. 

The  theoretical  velocity  at  the  centre  is  v  20,  and  the  discharge 
assuming  this  velocity  for  the  whole  section  is 

"6  x  -04  x  v/2#  -  '024  J2g  cubic  metres. 

The  actual  discharge,  on  the  above  assumption,  through  any 
horizontal  filament  of  thickness  dh,  and  depth  h,  is 

oQ  =  G'1549  xdfcx  J2ght 

and  the  total  discharge  is 

r  1-0775 
Q  -  01549  x/20  h*dh 

J  '9225 

-  '0241  J2g. 

The  theoretical  discharge,  taking  account  of  the  varying  heads 
is,  therefore,  1'004  times  the  discharge  calculated  on  the  assumption 
that  the  head  is  constant. 

As  the  head  is  increased  this  difference  diminishes,  and  when 
the  head  is  greater  than  5  times  the  depth  of  the  orifice,  is  very 
small  indeed. 

The  assumed  data  agrees  very  approximately  with  that  given 
in  Table  I  for  a  square  orifice,  where  the  value  of  ~k  is  given  as 
1-006. 


FLOW   THROUGH   ORIFICES  59 

This  partly  then,  explains  the  anomalous  values  of  k,  but  it 
cannot  be  looked  upon  as  a  complete  explanation. 

The  conditions  in  the  actual  jet  are  not  exactly  those  assumed, 
and  the  variation  of  velocity  normal  to  the  plane  of  the  section  is 
probably  much  more  complicated  than  here  assumed. 

As  Bazin  further  points  out,  it  is  probable  that,  in  jets  like 
those  from  the  square  orifice,  which,  as  will  be  seen  later  when  the 
form  of  the  jet  is  considered,  are  subject  to  considerable  deformation, 
the  divergence  of  some  of  the  filaments  gives  rise  to  pressures  less 
than  that  of  the  atmosphere. 

Bazin  has  attempted  to  demonstrate  this  experimentally,  and 
his  instrument,  Fig.  150,  registered  pressures  less  than  that  of  the 
atmosphere;  but  he  doubts  the  reliability  of  the  results,  and 
points  out  the  extreme  difficulty  of  satisfactorily  determining  the 
pressure  in  the  jet. 

That  the  inequality  of  the  velocity  of  the  filaments  is  the 
primary  cause,  receives  support  from  the  fact  that  for  the 
horizontal  orifice,  discharging  downwards,  the  coefficient  fc  is 
always  slightly  less  than  unity.  In  this  case,  in  any  horizontal 
section  below  the  orifice,  the  head  is  the  same  for  all  the  stream 
lines,  and  the  velocity  of  the  filaments  is  practically  constant. 
The  coefficient  of  velocity  is  never  less  than  '96,  so  that  the  loss 
due  to  the  internal  friction  of  the  liquid  is  very  small. 

40.  Distribution  of  velocity  in  the  plane  of  the  orifice. 
Bazin  has  examined   the  distribution  of  the  velocity  in   the 

various  sections  of  the  jet  by  means  of  a  fine  Pitot  tube  (see 
page  245).  In  the  plane  of  the  orifice  a  minimum  velocity 
occurs,  which  for  vertical  orifices  is  just  above  the  centre,  but  at  a 
little  distance  from  the  orifice  the  minimum  velocity  is  at  the  top 
of  the  jet. 

For  orifices  having  complete  contraction  Bazin  found  the 
minimum  velocity  to  be  '62  to  '64  \/2#H,  and  for  the  rectangular 
orifice,  with  lateral  contraction  suppressed,  0*69  \/2pH. 

As  the  distance  from  the  plane  of  the  orifice  increases,  the 
velocities  in  the  transverse  section  of  the  jets  from  horizontal 
orifices,  rapidly  become  uniform  throughout  the  transverse  section. 

For  vertical  orifices,  the  velocities  below  the  centre  of  the  jet 
are  greater  than  those  in  the  upper  part. 

41.  Pressure  in  the  plane  of  the  orifice. 

M.  Lagerjelm  stated  in  1826  that  if  a  vertical  tube  open  at 
both  ends  was  placed  with  its  lower  end  near  the  centre,  and  not 
perceptibly  below  the  plane  of  the  inner  edge  of  a  horizontal 


60  HYDRAULICS 

orifice  made  in  the  bottom  of  a  large  reservoir,  the  water  rose  in 
the  tube  to  a  height  equal  to  that  of  the  water  in  the  reservoir, 
that  is  the  pressure  at  the  centre  of  the  orifice  is  equal  to  the  head 
over  the  orifice  even  when  flow  is  taking  place. 

M.  Bazin  has  recently  repeated  this  experiment  and  found, 
that  the  water  in  the  tube  did  not  rise  to  the  level  of  the  water  in 
the  reservoir. 

If  Lager  j  elm's  statement  were  correct  it  would  follow  that  the 
velocity  at  the  centre  of  the  orifice  must  be  zero,  which  again  does 
not  agree  with  the  results  of  Bazin's  experiments  quoted  above. 

42.     Coefficient  of  discharge. 

The  discharge  per  second  from  an  orifice,  is  clearly  the  area 
of  the  jet  at  the  contracted  section  GrK  multiplied  by  the  mean 
velocity  through  this  section,  and  is  therefore, 

Q  =  c  .  k  .  a  \/2#H. 
Or,  calling  m  the  coefficient  of  discharge, 


This  coefficient  m  is  equal  to  the  product  c.Jc.  It  is  the  only 
coefficient  required  in  practical  problems  and  fortunately  it  can 
be  more  easily  determined  than  the  other  two  coefficients  c  and  Jc. 

Experimental  determination  of  the  coefficient  of  discharge. 
The  most  satisfactory  method  of  determining  the  coefficient  of 
discharge  of  orifices  is  to  measure  the  volume,  or  the  weight  of 
water,  discharged  under  a  given  head  in  a  known  time. 

The  coefficients  quoted  in  the  Tables  from  M.  Bazin*,  were 
determined  by  finding  accurately  the  time  required  to  fill  a  vessel 
of  known  capacity. 

The  coefficient  of  discharge  m,  has  been  determined  with 
a  great  degree  of  accuracy  for  sharp-edged  orifices,  by  Poncelet 
and  Lesbrost,  WeisbachJ,  Bazin  and  others  §.  In  Table  IY 
Bazin's  values  for  m  are  given. 

The  values  as  given  in  Tables  Y  and  YI  may  be  taken  as 
representative  of  the  best  experiments. 

For  vertical,  circular  and  square  orifices,  and  for  a  head  of 
about  3  feet  above  the  centre  of  the  orifice,  Mr  Hamilton  Smith, 
junr.  ||,  deduces  the  values  of  m  given  in  Table  YI. 

*  Annales  des  Fonts  et  Chaussees,  October,  1888. 

f  Floiv  through  Vertical  Orifices. 

J  Mechanics  of  Engineering. 

§  Experiments  upon  the  Contraction  of  the  Liquid  Vein.  Bazin  translated  by 
Trautwine. 

||  The  Flow  of  Water  through  Orifices  and  over  Weirs  and  through  open  Conduits 
and  Pipe*,  Hamilton  Smith,  junr.,  1886. 


FLOW  THROUGH   ORIFICES 

TABLE  Y. 


61 


Experimenter 

Particulars  of  orifice 

Coefficient  of 
discharge  m 

Bazin 

Poncelet  and 

Vertical  square  orifice  side  of  square  0*6562  ft. 

0-606 

Lesbros 

»                    »                    »i                    „ 

0-605 

Bazin 

Vertical  Rectangular  orifice  -656  ft.  high  x  2'624 
ft.  wide  with  side  contraction  suppressed 
Vertical  circular  orifice  0*6562  ft.  diameter 

0-627 
0-598 

„ 

Horizontal        „                    „                    } 

0-6035 

" 

0-3281 

0-6063 

TABLE  VI. 

Circular  orifices. 


Diameter  of 
orifice  in  ft. 

m 

0-0197 
0-627 

0-0295 
0-617 

0-039 
0-611 

0-0492 
0-606 

0-0984 
0-603 

0-164 
0-600 

0-328 
0-599 

0-6562 
0-598 

0-9843 
0-597 

Square  orifices. 


Side  of  square 
in  feet 

m 

0-0197 
0-631 

0-0492 
0-612 

0-0984 
0-607 

0-197 
0-605 

0-5906 
0-604 

0-9843 
0-603 

TABLE  VII. 

Table  showing  coefficients  of  discharge  for  square  and  rect- 
angular orifices  as  determined  by  Poncelet  and  Lesbros. 


Width  of  orifice  -6562  feet 

Width  of  orifice 
1  -968  feet 

Head  of  water 

above  the  top 

of  the  orifice 

Depth  of  orifice  in  feet 

in  feet 

•0328 

•0656 

•0984 

•1640 

•3287 

•6562 

•0656 

•6562 

•0328 

•701 

•660 

•630 

•607 

•0656 

•694 

•659 

•634 

•615 

•596 

•572 

•643 

•1312 

•683 

•658 

•640 

•623 

•603 

•582 

•642 

•595 

•2624 

•670 

•656 

•638 

•629 

•610 

•589 

•640 

•601 

•3937 

•663 

•653 

•636 

•630 

•612 

•593 

•638 

•603 

•6562 

•655 

•648 

•633 

•630 

•615 

•598 

•635 

•605 

1-640 

•642 

•638 

•630 

•627 

•617 

•604 

•630 

•607 

3-281 

•632 

•633 

•628 

•626     i  -615 

•605 

•626 

•605 

4-921 

•615 

•619 

•620 

•620 

•611 

•602 

•623 

•602 

6-562 

•611 

•612 

•612 

•613 

•607 

•601 

•620 

•602 

9-843 

•609 

•610 

•608 

•606 

•603 

•601 

•615 

•601 

62  HYDRAULICS 

The  heads  for  which  Bazin  determined  the  coefficients  in 
Tables  IY  and  V  varied  only  from  2'6  to  3*3  feet,  but,  as  will  be 
seen  from  Table  VII,  deduced  from  results  given  by  Poncelet  and 
Lesbros*  in  their  classical  work,  when  the  variation  of  head  is  not 
small,  the  coefficients  for  rectangular  and  square  orifices  vary 
considerably  with  the  head. 

43.  Effect  of  suppressed  contraction  on  the  coefficient 
of  discharge. 

Sharp-edged  orifice.  When  some  part  of  the  contraction  of  a 
transverse  section  of  a  jet  issuing  from  an  orifice  is  suppressed, 
the  cross  sectional  area  of  the  jet  can  only  be  obtained  with 
difficulty. 

The  coefficient  of  discharge  can,  however,  be  easily  obtained, 
as  before,  by  determining  the  discharge  in  a  given  time.  The 
most  complete  and  accurate  experiments  on  the  effect  of  contrac- 
tion are  those  of  Lesbros,  some  of  the  results  of  which  are  quoted 
in  Table  VIII.  The  coefficient  is  most  constant  for  square  or 
rectangular  orifices  when  the  lateral  contraction  is  suppressed.  The 
reason  being,  that  whatever  the  head,  the  variation  in  the  section 
of  the  jet  is  confined  to  the  top  and  bottom  of  the  orifice,  the 
width  of  the  stream  remaining  constant,  and  therefore  in  a  greater 
part  of  the  transverse  section  the  stream  lines  are  normal  to  the 
plane  of  the  orifice. 

According  to  Bidone,  if  x  is  the  fraction  of  the  periphery  of  a 
sharp-edged  orifice  upon  which  the  contraction  is  suppressed,  and 
m  the  coefficient  of  discharge  when  the  contraction  is  complete, 
then  the  coefficient  for  incomplete  contraction  is, 


for  rectangular  orifices,  and 

mi  =  m  (1 
for  circular  orifices. 

Bidone's   formulae    give    results    agreeing    fairly    well    with 
Lesbros'  experiments. 

His  formulae  are,  however,  unsatisfactory  when  x  approaches 
unity,  as  in  that  case  mi  should  be  nearly  unity. 

If  the  form  of  the  formula  is  preserved,  and  m  taken  as  '606, 
for  mi  to  be  unity  it  would  require  to  have  the  value, 


For  accurate  measurements,  either  orifices  with  perfect  con- 
traction or,  if  possible,  rectangular  or  square  orifices  with  the 
lateral  contraction  completely  suppressed,  should  be  used.  It  will 

*  Experiences  hydrauliques  sur  les  lois  de  Vecoulement  de  Veau  a  travers  les 
orifices,  etc.,  1832.  Poncelet  and  Lesbros. 


FLOW  THROUGH   ORIFICES 


63 


generally  be  necessary  to  calibrate  the  orifice  for  various  heads, 
but  as  shown  above  the  coefficient  for  the  latter  kind  is  more 
likely  to  be  constant. 

TABLE  VIII. 

Table  showing  the  effect  of  suppressing  the  contraction  on  the 
coefficient  of  discharge.     Lesbros*. 

Square  vertical  orifice  0'656  feet  square. 


Head  of  water 
above  the  upper 
edge  of  the  orifice 

Sharp-edged 

Side  con- 
traction 
suppressed 

Contraction 
suppressed  at 
the  lower  edge 

Contraction 
suppressed  at 
the  lower  and 
side  edges 

0-6562 

0-572 

0-599 

0-1640 

0-585 

0-631 

0-608 

0-3281 

0-592 

0-631 

0-615 

0-6562 

0-598 

0-632 

0-621 

0-708 

1-640 

0-603 

0-631 

0-623 

0-680 

3-281 

0-605 

0-628 

0-624 

0-676 

4-921 

0-602 

0-627 

0-624 

0-672 

6-562 

0-601 

0-626 

0-619 

0-668 

9-843 

0-601 

0-624 

0-614 

0-665 

circular  orifice. 
*     .4 


44.     The  form  of  the  jet  from  sharp-edged  orifices. 

From  a  circular  orifice  the  jet  emerges  like  a  cylindrical  rod 
and  retains  a  form  nearly  cylindrical  for  some  distance  from  the 
orifice. 

Fig.  51  shows  three  sections  of  a  jet  from  a  vertical  circular 
orifice  at  varying  distances  from  the       Fig.  51.    Section  of  jet  from 
orifice,  as  given  by  M.  Bazin. 

The  flow  from  square  orifices  is 
accompanied  by  an  interesting  and 
curious  phenomenon  called  the  in- 
version of  the  jet. 

At  a  very  small  distance  from 
the  orifice  the  section  becomes  as 
shown  in  Fig.  52.  The  sides  of  the 
jet  are  concave  and  the  corners  are 
cut  off  by  concave  sections.  The  Figs.  52—54.  Section  of  jet  from 
section  then  becomes  octagonal  as  in  84uare  orifice" 

Fig.  53  and  afterwards  takes  the  form  of  a  square  with  concave 
sides  and  rounded  corners,  the  diagonals  of  the  square  being 
perpendicular  to  the  sides  of  the  orifice,  Fig.  54. 

*  Experiments  hydrauliques  sur  les  lois  de  Vecoulement  de  Veau. 


64 


HYDRAULICS 


45.     Large  orifices. 

Table  YII  shows  very  clearly  that  if  the  depth  of  a  vertical  orifice 
is  not  small  compared  with  the  head,  the  coefficient  of  discharge 
varies  very  considerably  with  the  head,  and  in  the  discussion  of 
the  coefficient  of  velocity  k,  it  has  already  been  shown  that  the 
distribution  of  velocity  in  jets  issuing  from  such  orifices  is  not 
uniform.  As  the  jet  moves  through  a  large  orifice  the  stream 
lines  are  not  normal  to  its  plane,  but  at  some  section  of  the  stream 
very  near  to  the  orifice  they  are  practically  normal. 

If  now  it  is  assumed  that  the  pressure  is  constant  and  equal  to 
the  atmospheric  pressure  and  that  the  shape  of  this  section  is 
known,  the  discharge  through  it  can  be  calculated. 

Rectangular  orifice.  Let  efgh,  Fig.  55,  be  the  section  by  a 
vertical  plane  EF  of  the  stream  issuing  from  a  vertical  rectangular 
orifice.  Let  the  crest  E  of  the  stream  be  at  a  depth  h  below 
the  free  surface  of  the  water  in  the  vessel  and  the  under  edge 
F  at  a  depth  h2. 


^ 

e                  K 

f                  ff 

\  J 

^x 

i 

r      B 


Fig.  55. 

At  any  depth  h,  since  the  pressure  is  assumed  constant  in  the 
section,  the  fall  of  free  level  is  h,  and  the  velocity  of  flow  through 
the  strip  of  width  dh  is  therefore,  kv2gh,  and  the  discharge  is 


If  Jc  be  assumed  constant  for  all  the  filaments  the  total  discharge 
in  cubic  feet  per  second  is 

Q  - 

Here  at  once  a  difficulty  is  met  with.  The  dimensions  h0,  hi 
and  b  cannot  easily  be  determined,  and  experiment  shows  that 
they  vary  with  the  head  of  water  over  the  orifice,  and  that  they 
cannot  therefore  be  written  as  fractions  of  H0,  Hi,  and  B. 


FLOW   THROUGH    ORIFICES 


65 


By  replacing  h0,  h^  and  b  by  H0,  Hj  and  B  an  empirical 
formula  of  the  same  form  is  obtained  which,  by  introducing  a 
coefficient  c,  can  be  made  to  agree  with  experiments.  Then 


or  replacing  f  c  by  n, 


(1). 


The  coefficient  n  varies  with  the  head  H0,  and  for  any  orifice 
the  simpler  formula 

Q=m.a.N/2<7H  ..............................  (2), 

a  being  the  area  of  the  orifice  and  H  the  head  at  the  centre, 
can  be  used  with  equal  confidence,  for  if  n  is  known  for  the 
particular  orifice  for  various  values  of  H0,  ra  will  also  be  known. 

From  Table  VII  probable  values  of  m  for  any  large  sharp- 
edged  rectangular  orifices  can  be  interpolated. 

Rectangular  sluices.  If  the  lower  edge  of  a  sluice  opening  is 
some  distance  above  the  bottom  of  the  channel  the  discharge 
through  it  will  be  practically  the  same  as  through  a  sharp-edged 
orifice,  but  if  it  is  flush  with  the  bottom  of  the  channel,  the 
contraction  at  this  edge  is  suppressed  and  the  coefficient  of 
discharge  will  be  slightly  greater  as  shown  in  Table  VIII. 

46.     Drowned  orifices. 

When  an  orifice  is  submerged  as  in  Fig.  56  and  the  water  in 
the  up-stream  tank  or  reservoir  is  moving  so  slowly  that  its  velocity 
may  be  neglected,  the  head  causing  velocity  of  flow  through  any 
filament  is  equal  to  the  difference  of  the  up-  and  down-stream 
levels.  Let  H  be  the  difference  of  level  of  the  water  on  the  twa 
sides  of  the  orifice. 


Fig.  56. 


L.  H. 


66 


HYDRAULICS 


Consider  any  stream  line  FE  which  passes  through  the  orifice 
at  E.  The  pressure  head  at  E  is  equal  to  h2,  the  depth  of  E  below 
the  down-stream  level.  If  then  at  F  the  velocity  is  zero, 


or 


or  taking  a  coefficient  of  velocity 


which,  since  H  is  constant,  is  the  same  for  all  filaments  of  the 
orifice. 

If  the  coefficient  of  contraction  is  c  the  whole  discharge  through 
the  orifice  is  then 

Q  = 


47.    Partially  drowned  orifice. 

If  the  orifice  is  partially  drowned,  as  in 
Fig.  57,  the  discharge  may  be  considered  in 
two  parts.  Through  the  upper  part  AC  the 
discharge,  using  (2)  section  45,  is 


and  through  the  lower  part  BC 


Hx.  B 

48.    Velocity  of  approach. 

It  is  of  interest  to  consider  the  effect  of  the 
water  approaching  an  orifice  having  what  is  Fl&-  57- 

called  a  velocity  of  approach,  which  will  be  equal  to  the  velocity 
of  the  water  in  the  stream  above  the  orifice. 

In  Fig.  56  let  the  water  at  F  approaching  the  drowned  orifice 
have  a  velocity  VL 

Bernoulli's  equation  for  the  stream  line  drawn  is  then 

V     .       *     V 

^     '  "H  fl<>  —  li  ~l~  ^        • 


and 

which  is  again  constant  for  all  filaments  of  the  orifice. 
Then  Q  =  m.c 


SUDDEN   ENLARGEMENT  OF  A  STREAM  67 

49.  Effect  of  velocity  of  approach  on  the  discharge 
through  a  large  rectangular  orifice. 

If  the  water  approaching  the  large  orifice,  Fig.  55,  has 
a  velocity  of  approach  v1}  Bernoulli's  equation  for  the  stream  line 
passing  through  the  strip  at  depth  h,  will  be 

fe+*-fe  +*+*', 

W      2ff  .    ID  2g' 

pa  being  the  atmospheric  pressure,  or  putting  in  a  coefficient  of 
velocity, 


The  discharge  through  the  orifice  is  now, 


50.     Coefficient  of  resistance. 

In  connection  with  the  flow  through  orifices,  and  hydraulic 
plant  generally,  the  term  "  coefficient  of  resistance  "  is  frequently 
used.  Two  meanings  have  been  attached  to  the  term.  Some- 
times it  is  defined  as  the  ratio  of  the  head  lost  in  a  hydraulic 
system  to  the  effective  head,  and  sometimes  as  the  ratio  of  the 
head  lost  to  the  total  head  available.  According  to  the  latter 
method,  if  H  is  the  total  head  available  and  hf  the  head  lost, 
the  coefficient  of  resistance  is 


51.     Sudden  enlargement  of  a  current  of  water. 

It  seems  reasonable  to  proceed  from  the  consideration  of  flow 
through  orifices  to  that  of  the  flow  through  mouthpieces,  but 
before  doing  so  it  is  desirable  that  the  effect  of  a  sudden 
enlargement  of  a  stream  should  be  considered. 

Suppose  for  simplicity  that  a  pipe  as 
in  Fig.  58  is  suddenly  enlarged,  and  that 
there  is  a  continuous  sinuous  flow  along 
the  pipe.  (See  section  284.) 

At  the  enlargement  of  the  pipe,  the 
stream  suddenly  enlarges,  and,  as  shown 
in  the  figure,  in  the  corners  of  the  large 
pipe  it  may  be  assumed  that  eddy  motions 
are  set  up  which  cause  a  loss  of  energy. 

5—2 


68  HYDRAULICS 

Consider  two  sections  aa  and  dd  at  such  a  distance  from  66 
that  the  flow  is  steady. 

Then,  the  total  head  at  dd  equals  the  total  head  at  aa  minus 
the  loss  of  head  between  aa  and  dd,  or  if  h  is  the  loss  of  head  due 
to  shock,  then 

&+v/.=p«+vji+h. 

w      2g      w      2g 

Let  A«  and  A^  be  the  area  at  aa  and  dd  respectively. 
Since  the  flow  past  aa  equals  that  past  dd, 


Then,   assuming  that  each  filament   of  fluid   at   aa  has   the 
velocity  va,  and  vd  at  dd,  the  momentum  of  the  quantity  of  water 

nt\ 

which  passes  aa  in  unit  time  is  equal  to  —  A^Va,  and  the  momentum 

y 
of  the  water  that  passes  dd  is 


the  momentum  of  a  mass  of  M  pounds  moving  with  a  velocity 
v  feet  per  second  being  M.V  pounds  feet. 
The  change  of  momentum  is  therefore, 


The  forces  acting  on  the  water  between  aa  and  dd  to  produce 
this  change  of  momentum,  are 

paAa  acting  on  aa,  pd^.d  acting  on  dd, 

and,  if  p  is  the  mean  pressure  per  unit  area  on  the  annular  ring 
66,  an  additional  force  p(A.d  —  Aa). 

There  is  considerable  doubt  as  to  what  is  the  magnitude  of  the 
pressure  p,  but  it  is  generally  assumed  that  it  is  equal  to  pa,  for 
the  following  reason. 

The  water  in  the  enlarged  portion  of  the  pipe  may  be  looked 
upon  as  divided  into  two  parts,  the  one  part  having  a  motion  of 
translation,  while  the  other  part,  which  is  in  contact  with  the 
annular  ring,  is  practically  at  rest.  (See  section  284.) 

If  this  assumption  is  correct,  then  it  is  to  be  expected  that  the 
pressure  throughout  this  still  water  will  be  practically  equal  at  all 
points  and  in  all  directions,  and  must  be  equal  to  the  pressure  in 
the  stream  at  the  section  66,  or  the  pressure  p  is  equal  to  pa. 

Therefore 

v 

-  Aa)  -  paA^  =  W 


y 
from  which         (pd  -  pa)  Ad  =  w  —  —  (va  -  vd)  ; 


SUDDEN    ENLARGEMENT   OF   A   STREAM 


69 


and  since 
therefore 


w       w        g         g 


v 

Adding  -?j-    to   both   sides   of    the   equation   and    separating 


„.   2 

-  into  two  parts, 


w      2g      w      2g  ""        20       ' 
or  /&  the  loss  of  head  due  to  shock  is  equal  to 


According  to  St  Venant  this  quantity  should  be  increased  by 
an  amount  equal  to  ~  ~-  ,  but  this  correction  is  so  small  that  as 
a  rule  it  can  be  neglected. 

52.     Sudden  contraction  of  a  current  of  water. 

Suppose  a  pipe  partially  closed  by  means  of  a  diaphragm  as  in 
Fig.  59. 

As  the  stream  approaches  the  diaphragm 
—  which  is  supposed  to  be  sharp-edged  — 
it  contracts  in  a  similar  way  to  the  stream 
passing  through  an  orifice  on  the  side  of 
a  vessel,  so  that  the  minimum  cross  sec- 
tional area  of  the  flow  will  be  less  than  the 
area  of  the  orifice. 

The  loss  of  head  due  to  this  contraction,  or  due  to  passing 
through  the  orifice  is  small,  as  seen  in  section  39,  but  due  to 
the  sudden  enlargement  of  the  stream  to  fill  the  pipe  again,  there 
is  a  considerable  loss  of  head. 

Let  A  be  the  area  of  the  pipe  and  a  of  the  orifice,  and  let  c  be 
the  coefficient  of  contraction  at  the  orifice. 

Then  the  area  of  the  stream  at  the  contracted  section  is  ca,  and, 
therefore,  the  loss  of  head  due  to  shock 


Fig.  59. 


\  ca 


70  HYDRAULICS 

If  the  pipe  simply  diminishes  in  diameter  as  in  Fig.  58,  the 
section  of  the  stream  enlarges  from  the  contracted  area  ca  to  fill 
the  pipe  of  area  a,  therefore  the  loss  of  head  in  this  case  is 


Or  making  St  Venant  correction 

'       .....................  * 


Value  of  the  coefficient  c.  The  mean  value  of  c  for  a  sharp-edged 
circular  orifice  is,  as  seen  in  Table  IY,  about  0'6,  and  this  may  be 
taken  as  the  coefficient  of  contraction  in  this  formula. 

Substituting  this  value  in  equation   (1)  the  loss  of  head  is 


found  to  be  --  ,  and  in  equation  (2),  -  -  ,  v  being  the  velocity  in 
*9    '  ^9 

C\*  K.     2 

the  small  pipe.     It  may  be  taken  therefore   as  -~  —  .     Further 
experiments  are  required  before  a  correct  value  can  be  assigned. 

53.  Loss  of  head  due  to  sharp-edged  entrance  into  a  pipe 
or  mouthpiece. 

When  water  enters  a  pipe  or  mouthpiece  from  a  vessel  through 
a  sharp-edged  entrance,  as  in  Fig.  61,  there  is  first  a  contraction,  and 
then  an  enlargement,  as  in  the  second  case  considered  in  section  52. 

The  loss  of  head  may  be,  therefore,  taken  as  approximately  —  ~  — 

O'SOSv2 
and  this  agrees  with  the  experimental  value  of  —  ~  --  given  by 

Weisbach. 

This  value  is  probably  too  high  for  small  pipes  and  too  low  for 
large  pipes*. 

54,  Mouthpieces. 

If  an  orifice  is  provided  with  a  short  pipe  or  mouthpiece,  through 
which  the  liquid  can  flow,  the  discharge  may  be  very  different 
from  that  of  a  sharp-edged  orifice,  the  difference  depending  upon 
the  length  and  form  of  the  mouthpiece.  If  the  orifice  is  cylindrical 
as  shown  in  Fig.  60,  being  sharp  at  the  inner  edge,  and  so  short 
that  the  stream  after  converging  at  the  inner  edge  clears  the 
outer  edge,  it  behaves  as  a  sharp-edged  orifice. 

Short  external  cylindrical  mouthpieces.  If  the  mouthpiece  is 
cylindrical  as  ABFE,  Fig.  61,  having  a  sharp  edge  at  AB  and 
a  length  of  from  one  and  a  half  to  twice  its  diameter,  the  jet 

*  See  M.  Bazin,  Experiences  nouvelles  sur  la  distribution  des  vitesses  dans 
les  tuyaux. 


FLOW   THROUGH   MOUTHPIECES 


71 


contracts  to  CD,  and  then  expands  to  fill  the  pipe,  so  that  at  EF 
it  discharges  full  bore,  and  the  coefficient  of  contraction  is  then 
unity.  Experiment  shows,  that  the  coefficient  of  discharge  is 


Fig.  60. 


Fig.  61. 


from  0'80  to  0*85,  the  coefficient  diminishing  with  the  diameter 
of  the  tube.  The  coefficient  of  contraction  being  unity,  the 
coefficients  of  velocity  and  discharge  are  equal.  Good  mean 
values,  according  to  Weisbach,  are  0*815  for  cylindrical  tubes, 
and  0*819  for  tubes  of  prismatic  form. 

These  coefficients  agree  with  those  determined  on  the  assump- 
tion that  the  only  head  lost  in  the  mouthpiece  is  that  due  to 
sudden  enlargement,  and  is 

0*5 


v  being  the  velocity  of  discharge  at  EF. 

Applying  Bernoulli's  theorem  to  the  sections  CD  and  EF,  and 

'5v2 

taking  into  account  the  loss  of  head  of  -~— ,  and  pa  as  the  atmo- 
spheric pressure, 

PCD    ,    ^CD2       Pa   ,     ^    ,    '§V*  _  TT       Pa 
r  —p: —  = r  p: P  ~p: —  —  -CL  T         , 

w        2g      w      2g      2g  w 


or 


Therefore 


-66  x 


and  v 

The  area  of  the  jet  at  EF  is  a,  and  therefore,  the  discharge 
per  second  is 


Or  m,  the  coefficient  of  discharge,  is  0*812. 
The  pressure  head  at  the  section  CD.     Taking  the  area  at  CD 
as  0*606  the  area  at  EF, 


72  HYDRAULICS 

Therefore  C->.fc  +  "^  , 

w       w        Zg          Zg         w         Zg 

or  the  pressure  at  C  is  less  than  the  atmospheric  pressure. 

If  a  pipe  be  attached  to  the  mouthpiece,  as  in  Fig.  61,  and  the 
lower  end  dipped  in  water,  the  water  should  rise  to  a  height  of  about 

1'22-y2 

—  2  —  feet  above  the  water  in  the  vessel. 

55.    Borda's  mouthpiece. 

A  short  cylindrical  mouthpiece  projecting  into  the  vessel,  as  in 
Fig.  62,  is  called  a  Borda's  mouthpiece,  and  is  of  interest,  as  the 
coefficient  of  discharge  upon  certain  assumptions  can  be  readily 
calculated.  Let  the  mouthpiece  be  so  short 
that  the  jet  issuing  at  EF  falls  clear  of  GrH. 
The  orifice  projecting  into  the  liquid  has 
the  effect  of  keeping  the  liquid  in  contact 
with  the  face  AD  practically  at  rest,  and 
at  all  points  on  it  except  the  area  EF  the 
hydrostatic  pressure  will,  therefore,  simply 
depend  upon  the  depth  below  the  free 

surface  AB.  Imagine  the  mouthpiece  produced  to  meet  the 
face  BC  in  the  area  IK.  Then  the  hydrostatic  pressure  on  AD, 
neglecting  EF,  will  be  equal  to  the  hydrostatic  pressure  on  BC, 
neglecting  IK. 

Again,  BC  is  far  enough  away  from  EF  to  assume  that  the 
pressure  upon  it  follows  the  hydrostatic  law. 

The  hydrostatic  pressure  on  IK,  therefore,  is  the  force  which 
gives  momentum  to  the  water  escaping  through  the  orifice,  over- 
comes the  pressure  on  EF,  and  the  resistance  of  the  mouthpiece. 

Let  H  be  the  depth  of  the  centre  of  the  orifice  below  the  free 
surface  and  p  the  atmospheric  pressure.  Neglecting  frictional 
resistances,  the  velocity  of  flow  v,  through  the  orifice,  is  X/20H. 

Let  a  be  the  area  of  the  orifice  and  <o  the  area  of  the  transverse 
section  of  the  jet.  The  discharge  per  second  will  be  w  .  w  \/2gH  Ibs. 

The  hydrostatic  pressure  on  IK  is 

pa  +  waH.  Ibs. 

The  hydrostatic  pressure  on  EF  is  pa  Ibs. 

The  momentum  given  to  the  issuing  water  per  second,  is 


Therefore  pa  +  —  w  2gR  =  pa  +  waK, 

and  w  =   a. 


FLOW  THROUGH   MOUTHPIECES  73 

The  coefficient  of  contraction  is  then,  in  this  case,  equal  to 
one  half. 

Experiments  by  Borda  and  others,  show  that  this  result  is 
justified,  the  experimental  coefficient  being  slightly  greater 
than  J. 

56.     Conical  mouthpieces  and  nozzles. 

These  are  either  convergent  as  in  Fig.  63,  or  divergent  as  in 
Fig.  64. 


Fig.  63. 


Fig.  64. 


Calling  the  diameter  of  the  mouthpiece  the  diameter  at  the 
outlet,  a  divergent  tube  gives  a  less,  and  a  convergent 
tube  a  greater  discharge  than  a  cylindrical  tube  of  the 
same  diameter. 

Experiments  show  that  the  maximum  discharge  for  a 
convergent  tube  is  obtained  when  the  angle  of  the  cone 
is  from  12  to  13J  degrees,  and  it  is  then  0'94 .  a .  J2gh. 
If,  instead  of  making  the  convergent  mouthpiece  conical, 
its  sides  are  curved  as  in  Fig.  65,  so  that  it  follows  as 
near  as  possible  the  natural  form  of  the  stream  lines,  the 
coefficient  of  discharge  may,  with  high  heads,  approxi- 
mate very  nearly  to  unity. 

Weisbach*,  using  the  method  described  on  page  55 
to    determine    the   velocity  of    flow,   obtained,  for  this 
mouthpiece,  the  following  values  of  k.     Since  the  mouth-    Fig<  65 
piece  discharges  full  the  coefficients  of  velocity  k  and 
discharge  m  are  practically  equal. 


Head  in  feet 
k  and  m 


0-66 
•959 


1-64 
•967 


11-48 
•975 


55-8 
•994 


338 
•994 


According  to  Freeman  t,  the  fire-hose  nozzle  shown  in  Fig.  66 
has  a  coefficient  of  velocity  of  "977. 


*  Mechanics  of  Engineering. 

f  Transactions  Am.  Soc.  C.E.,  Vol.  xxi. 


74  HYDRAULICS 

If  the  mouthpiece  is  first  made  convergent,  and  then  divergent, 


Fig.  66. 

as  in  Fig.  67,  the  divergence  being  sufficiently  gradual  for  the 
stream  lines  to  remain  in  contact  with  the  tube,  the  coefficient  of 
contraction  is  unity  and  there  is  but  a 
small  loss  of  head.  The  velocity  of  efflux 
from  EF  is  then  nearly  equal  to  \/2grH 
and  the  discharge  is  m .  a .  v/2011,  a  being 
the  area  of  EF,  and  the  coefficient  m 
approximates  to  unity. 

It  would  appear,  that  the  discharge 
could  be  increased  indefinitely  by  length- 
ening the  divergent  part  of  the  tube  and 


Fig.  67. 


thus  increasing  a,  but  as  the  length  increases,  the  velocity 
decreases  due  to  the  friction  of  the  sides  of  the  tube,  and  further, 
as  the  discharge  increases,  the  velocity  through  the  contracted 
section  CD  increases,  and  the  pressure  head  at  CD  consequently 
falls. 

Calling  pa  the  atmospheric  pressure,  pi  the  pressure  at  CD, 
and  Vi  the  velocity  at  CD,  then 


w      2g 


w 


and 


Fi_j£  +  /^a_  "1 

w  w     2g ' 

Pa 


If    -  is  greater  than  H  +  —  ,  px  becomes  negative. 


As  pointed  out,  however,  in  connection  with  Froude's  apparatus, 
page  43,  if  continuity  is  to  be  maintained,  the  pressure  cannot  be 
negative,  and  in  reality,  if  water  is  the  fluid,  it  cannot  be  less 
than  3-  the  atmospheric  pressure,  due  to  the  separation  of  the  air 
from  the  water.  The  velocity  Vi  cannot,  therefore,  be  increased 
indefinitely. 


FLOW   THROUGH   MOUTHPIECES 


75 


Assuming  the  pressure  can  just  become  zero,  and  taking  the 
atmospheric  pressure  as  equivalent  to  a  head  of  34  ft.  of  water,  the 
maximum  possible  velocity,  is 


and  the  maximum  ratio  of  the  area  of  EF  to  CD  is 


34ft. 
H    ' 

Practically,  the  maximum  value  of  Vi  may  be  taken  as 


and  the  maximum  ratio  of  EF  to  CD  as 


24  ft. 

H    ' 


The  maximum  discharge  is 


a  J2g  (H  +  24) 
=  m. y  v ' 


The  ratio  given  of  EF  to  CD  may  be  taken  as  the  maximum 
ratio  between  the  area  of  a  pipe  and  the  throat  of  a  Yenturi  meter 
to  be  used  in  the  pipe. 

5  7.  Flow  through  orifices  and  mouthpieces  under  constant 
pressure. 

The  head  of  water  causing  flow  through  an  orifice  may  be 
produced  by  a  pump  or  other  mechanical  means,  and  the  discharge 
may  take  place  into  a  vessel,  such  as  the  condenser  of  a  steam 
engine,  in  which  the  pressure  is  less  than  that  of  the  atmosphere. 

For  example,  suppose  water  to  be  discharged  from  a  cylinder 
A,  into  a  vessel  B,  Fig.  68,  through 
an  orifice  or  mouthpiece  by  means 
of  a  piston  loaded  with  P  Ibs.,  and 
let  the  pressure  per  sq.  foot  in  B 
be  po  Ibs. 

Let  the  area  of  the  piston  be 
A  square  feet.  Let  h  be  the  height 
of  the  water  in  the  cylinder  above 
the  centre  of  the  orifice  and  h0  of 
the  water  in  the  vessel  B.  The 


Fig.  68. 


theoretical  effective  head  forcing  water  through  the  orifice  may 
be  written 


H=  r- 

A.W 


w 


76  HYDRAULICS 

If  P  is  large  h0  and  h  will  generally  be  negligible. 
At  the  orifice  the  pressure  head  is  h0  +  —  ,  and  therefore  for 
any  stream  line  through  the  orifice,  if  there  is  no  friction, 

^+h+p°-  p  +h 

d       ~  /t/O  T  —       .  Til/ 

2g  w      A.W 

tf       P       ,      ,       p, 

or  fr-  =  -7  —  +  h-ho  —  —  . 

2g     Aw  w 

The  actual  velocity  will  be  less  than  vt  due  to  friction,  and  if  Jc 
is  a  coefficient  of  velocity,  the  velocity  is  then 


and  the  discharge  is          Q  =  m  .  a 

In  practical  examples  the  cylinder  and  the  vessel  will  generally 
be  connected  by  a  short  pipe,  for  which  the  coefficient  of  velocity 
will  depend  upon  the  length. 

If  it  is  only  a  few  feet  long  the  principal  loss  of  head  will  be 
at  the  entrance  to  the  pipe,  and  the  coefficient  of  discharge  will 
probably  vary  between  0'65  and  0*85. 

The  effect  of  lengthening  the  pipe  will  be  understood  after  the 
chapter  on  flow  through  pipes  has  been  read. 

Example.     Water  is  discharged  from  a  pump  into  a  condenser  in  which  the 
pressure  is  3  Ibs.  per  sq.  inch  through  a  short  pipe  3  inches  diameter. 
The  pressure  in  the  pump  is  20  Ibs.  per  sq.  inch. 

Find  the  discharge  into  the  condenser,  taking  the  coefficient  of  discharge  0*75. 
The  effective  head  is 

20x144      3x144 
:     62-4  62-4 

=  39-2  feet. 

Therefore,      Q  =  -75  x  -7854  x  ^  x  ^64-4  x  39-2  cubic  feet  per  sec. 
=  1-84  cubic  ft.  per  sec. 

58.    Time  of  emptying  a  tank  or  reservoir. 

Suppose  a  reservoir  to  have  a  sharp-edged  horizontal  orifice 
as  in  Fig.  44.  It  is  required  to  find  the  time  taken  to  empty 
the  reservoir. 

Let  the  area  of  the  horizontal  section  of  the  reservoir  at  any 
height  h  above  the  orifice  be  A  sq.  feet,  and  the  area  of  the 

orifice  a  sq.  feet,  and  let  the  ratio  —  be  sufficiently  large  that  the 

CL 

velocity  of  the  water  in  the  reservoir  may  be  neglected. 

When  the  surface  of  the  water  is  at  any  height  h  above  the 
orifice,  the  volume  which  flows  through  the  orifice  in  any  time  dt 
will  be  ma  \/2gh  .  dt. 


FLOW  THROUGH  MOUTHPIECES  77 

The  amount  dh  by  which  the  surface  of  water  in  the  reservoir 
falls  in  the  time  dtis 


or  d£  = 

ma 


The  time  for  the  water  to  fall  from  a  height  H  to  Hj  is 
(H      Adh  1       /•=  Adh 

I/  I  ~       '  .____ y_      —        I 

/  HJ  ma  \/2gh     a  *J2g  J  H,  mh^  ' 

If  A  is  constant,  and  m  is  assumed  constant,  the  time  required 
for  the  surface  to  fall  from  a  height  H  to  H!  above  the  orifice  is 

[H  Adh 


ma 

and  the  time  to  empty  the  vessel  is 

2A 


IcT  ' 

ma  v  Zg 

or  is  equal  to  twice  the  time  required  for  the  same  volume  of 
water  to  leave  the  vessel  under  a  constant  head  H. 

Time  of  emptying  a  lock  with  vertical  drowned  sluice.  Let  the 
water  in  the  lock  when  the  sluice  is  closed  be  at  a  height  H, 
Fig.  56,  above  the  down-stream  level. 

Then  the  time  required  is  that  necessary  to  reduce  the  level  in 
the  lock  by  an  amount  H. 

When  the  flow  is  taking  place,  let  x  be  the  height  of  the  water 
in  the  lock  at  any  instant  above  the  down-stream  water. 

Let  A  be  the  sectional  area  of  the  lock,  at  the  level  of  the 
water  in  the  lock,  a  the  area  of  the  sluice,  and  m  its  coefficient  of 
discharge. 

The  discharge  through  the  sluice  in  time  dt  is 

c)Q  =  m  .  a  \l2gx .  dt. 

If  dx  is  the  distance  the  surface  falls  in  the  lock  in  time  dt,  then 
Adx  =  ma  *j2gxdt, 

Adx 

or  dt  =  -       . —   .  . 

ma  v2<7#52 

To  reduce  the  level  by  an  amount  H, 

'H     Adx 


o  ma 


78  HYDRAULICS 

If  m  and  A  are  constant, 

ma  \/2g 

Example.  A  reservoir,  200  yards  long  and  150  yards  wide  at  the  bottom,  and 
having  side  slopes  of  1  to  1,  has  a  depth  of  water  in  it  of  25  feet.  A  short  pipe 
3  feet  diameter  is  used  to  draw  off  water  from  the  reservoir. 

Find  the  time  taken  for  the  water  in  the  reservoir  to  fall  10  feet.  The 
coefficient  of  discharge  for  the  pipe  is  0-7. 

When  the  water  has  a  depth  h  the  area  of  the  water  surface  is 

A  =  (600  +  27i)(450  +  27i). 
The  area  of  the  pipe  is  a  =  7'068  sq.  feet. 

Therefore          t= L-  —  /*  (600  +  2*)  (460  +  8*)  M 

0-70  J2g  .  7-068  J  15  hb 

=  -* a  f25  2  x  27000071*  + 1  x  21007**  +  f 7»*~| 

o9"O  l_15  _l 

=  ^ig  (610200  +  93800  +  3606) 

=  17,850  sees. 
=  4-95  hours. 

Example:  A  horizontal  boiler  6  feet  diameter  and  30  feet  long  is  half  full  of 
water. 

Find  the  time  of  emptying  the  boiler  through  a  short  vertical  pipe  3  inches 
diameter  attached  to  the  bottom  of  the  boiler. 

The  pipe  may  be  taken  as  a  mouthpiece  discharging  full,  the  coefficient  of 
velocity  for  which  is  0*8. 

Let  r  be  the  radius  of  the  boiler. 

When  the  water  has  any  depth  h  above  the  bottom  of  the  boiler  the  area  A  is 


The  area  of  the  pipe  is  0-049  sq.  feet. 

™       , 

Therefore  t  = 


8  x  0-049  *j2g  J  o         ,Jk 

=  95-5  /    (2r-h$dh 
o 


=  63-8x9-5 
=  606  sees. 


EXAMPLES. 

(1)  Find  the  velocity  due  to  a  head  of  100  ft. 

(2)  Find  the  head  due  to  a  velocity  of  500  ft.  per  sec. 

(3)  Water  issues  vertically  from  an  orifice  under  a  head  of  40  ft.     To 
what  height  will  the  jet  rise,  if  the  coefficient  of  velocity  is  0'97  ? 

(4)  What  must  be  the  size  of  a  conoidal  orifice  to  discharge  10  c.  ft. 
per  second  under  a  head  of  100  ft.  ?     m  =  -925. 


FLOW  THKOUGH   ORIFICES   AND  MOUTHPIECES  79 

(5)  A  jet  3  in.  diameter  at  the  orifice  rises  vertically  50  ft.     Find  its 
diameter  at  25  ft.  above  the  orifice. 

(6)  An  orifice  1  sq.  ft.  in  area  discharges  18  c.  ft.  per  second  under  a 
head  of  9  ft.     Assuming  coefficient  of  velocity =0'98,  find  coefficient  of 
contraction. 

(7)  The  pressure  in  the  pump  cylinder  of  a  fire-engine  is  14,400  Ibs. 
per  sq.  ft. ;  assuming  the  resistance  of  the  valves,  hose,  and  nozzle  is  such 
that  the  coefficient  of  resistance  is  0'5,  find  the  velocity  of  discharge,  and 
the  height  to  which  the  jet  will  rise. 

(8)  The  pressure  in  the  hose  of  a  fire-engine  is  100  Ibs.  per  sq.  inch; 
the  jet  rises  to  a  height  of  150  ft.     Find  the  coefficient  of  velocity. 

(9)  A  horizontal  jet  issues  under  a  head  of  9  ft.     At  6  ft.  from  the 
orifice  it  has  fallen  vertically  15  ins.     Find  the  coefficient  of  velocity. 

(10)  Required  the  coefficient  of  resistance  corresponding  to  a  coefficient 
of  velocity =0-97. 

(11)  A  fluid  of  one  quarter  the  density  of  water  is  discharged  from  a 
vessel  in  which  the  pressure   is  50  Ibs.  per  sq.  in.   (absolute)  into  the 
atmosphere  where  the  pressure  is  15  Ibs.  per  sq.  in.     Find  the  velocity  of 
discharge. 

(12)  Find  the  diameter  of  a  circular  orifice  to  discharge  2000  c.  ft.  per 
hour,  under  a  head  of  6  ft.     Coefficient  of  discharge  0'60. 

(13)  A  cylindrical  cistern  contains  water  16  ft.  deep,  and  is  1  sq.  ft.  in 
cross  section.     On  opening  an  orifice  of  1  sq.  in.  in  the  bottom,  the  water 
level  fell  7  ft.  in  one  minute.     Find  the  coefficient  of  discharge. 

(14)  A  miner's  inch  is  defined  to  be  the  discharge  through  an  orifice  in 
a  vertical  plane  of  1  sq.  in.  area,  under  an  average  head  of  6|  ins.     Find 
the  supply  of  water  per  hour  in  gallons.     Coefficient  of  discharge  0'62. 

(15)  A  vessel  fitted  with  a  piston  of  12  sq.  ft.  area  discharges  water 
under  a  head  of  10  ft.   What  weight  placed  on  the  piston  would  double  the 
rate  of  discharge  ? 

(16)  An  orifice  12  inches  square  discharges  under  a  head  of  100  feet 
1*338  cubic  feet  per  second.     Taking  the  coefficient  of  velocity  at  0'97,  find 
the  coefficient  of  contraction. 

(17)  Find  the  discharge  per  minute  from  a  circular  orifice  1  inch 
diameter,  under  a  constant  pressure  of  34  Ibs.  per  sq.  inch,  taking  0'60  as 
the  coefficient  of  discharge. 

(18)  The  plunger  of  a  fire-engine  pump  of  one  quarter  of  a  sq.  ft.  in 
area  is  driven  by  a  force  of  9542  Ibs.  and  the  jet  is  observed  to  rise  to  a 
height  of  150  feet.     Find  the  coefficient  of  resistance  of  the  apparatus. 

(19)  An  orifice  3  feet  wide  and  2  feet  deep  has  12  feet  head  of  water 
above  its  centre  on  the  up-stream  side,  and  the  backwater  on  the  other 
side  is  at  the  level  of  the  centre  of  the  orifice.    Find  the  discharge  if 


80  HYDRAULICS 

(20)  Ten  c.  ft.  of  water  per  second  flow  through  a  pipe  of  1  sq.  ft.  area, 
which  suddenly  enlarges  to  4  sq.  ft.  area.     Taking  the  pressure  at  100  Ibs. 
per  sq.  ft.  in  the  smaller  part  of  the  pipe,  find  (1)  the  head  lost  in  shock, 
(2)  the  pressure  in  the  larger  part,  (3)  the  work  expended  in  forcing  the 
water  through  the  enlargement. 

(21)  A  pipe  of  3"  diameter  is  suddenly  enlarged  to  5"  diameter.     A  U 
tube  containing  mercury  is  connected  to  two  points,  one  on  each  side  of  the 
enlargement,  at  points  where  the  flow  is  steady.     Find  the  difference  in 
level  in  the  two  limbs  of  the  U  when  water  flows  at  the  rate  of  2  c.  ft.  per 
second  from  the  small  to  the  large  section  and  vice  versd.     The  specific 
gravity  of  mercury  is  13'6.     Lond.  Un. 

(22)  A  pipe  is  suddenly  enlarged  from  2£  inches  in  diameter  to  3£ 
inches  in  diameter.    Water  flows  through  these  two  pipes  from  the  smaller 
to  the  larger,  and  the  discharge  from  the  end  of  the  bigger  pipe  is  two 
gallons  per  second.     Find: — 

(a)  The  loss  of  head,  and  gain  of  pressure  head,  at  the  enlarge- 
ment. 

(b)  The  ratio  of  head  lost  to  velocity  head  in  small  pipe. 

(23)  The  head  and  tail  water  of  a  vertical- sided  lock  differ  in  level 
12  ft.     The  area  of  the  lock  basin  is  700  sq.  ft.     Find  the  time  of  emptying 
the  lock,  through  a  sluice  of  5  sq.  ft.  area,  with  a  coefficient  0'5.     The 
sluice  discharges  below  tail  water  level. 

(24)  A  tank  1200  sq.  ft.  in  area  discharges  through  an  orifice  1  sq.  ft. 
in  area.     Calculate  the  time  required  to  lower  the  level  in  the  tank  from 
50  ft.  to  25  ft.  above  the  orifice.     Coefficient  of  discharge  0'6. 

(25)  A  vertical-sided  lock  is  65  ft.  long  and  18  ft.  wide.     Lift  15  ft. 
Find  the  area  of  a  sluice  below  tail  water  to  empty  the  lock  in  5  minutes. 
Coefficient  0'6. 

(26)  A  reservoir  has  a  bottom  width  of  100  feet  and  a  length  of  250 
feet. 

The  sides  of  the  reservoir  are  vertical. 

The  reservoir  is  connected  to  a  second  reservoir  of  the  same  dimensions 
by  means  of  a  pipe  2  feet  diameter.  The  surface  of  the  water  in  the  first 
reservoir  is  17  feet  above  that  in  the  other.  The  pipe  is  below  the  surface 
of  the  water  in  both  reservoirs.  Find  the  time  taken  for  the  water  in  the 
two  reservoirs  to  become  level.  Coefficient  of  discharge  0'8. 

59.     Notches  and  Weirs. 

When  the  sides  of  an  orifice  are 
produced,  so  that'  they  extend  be- 
yond the  free  surface  of  the  water, 
as  in  Figs.  69  and  70,  it  is  called  a 
notch. 

Notches  are  generally  made  tri- 
angular or  rectangular  as  shown 
in  the  figures  and  are  largely  used 
for  gauging  the  flow  of  water.  Fig.  69.  Triangular  Notch. 


FLOW  OVER   WEIRS  81 

For  example,  if  the  flow  of  a  small  stream  is  required,  a  dam  is 
constructed  across  the  stream  and  the  water  allowed  to  pass 
through  a  notch  cut  in  a  board  or  metal  plate. 


-  -L---H 


mimr 


Fig.  70.     Rectangular  Notch. 

They  can  conveniently  be  used  for  measuring  the  compensation 
water  to  be  supplied  from  collecting  reservoirs,  and  also  to  gauge 
the  supply  of  water  to  water  wheels  and  turbines. 

The  term  weir  is  a  name  given  to  a  structure  used  to  dam  up 
a  stream  and  over  which  the  water  flows. 

The  conditions  of  flow  are  practically  the  same  as  through 
a  rectangular  notch,  and  hence  such  notches  are  generally  called 
weirs,  and  in  what  follows  the  latter  term  only  is  used.  The  top 
of  the  weir  corresponds  to  the  horizontal  edge  of  the  notch  and  is 
called  the  sill  of  the  weir. 

The  sheet  of  water  flowing  over  a  weir  or  through  a  notch  is 
generally  called  the  vein,  sheet,  or  nappe. 

The  shape  of  the  nappe  depends  upon  the  form  of  the  sill  and 
sides  of  the  weir,  the  height  of  the  sill  above  the  bottom  of  the 
up-stream  channel,  the  width  of  the  up-stream  channel,  and  the 
construction  of  the  channel  into  which  the  nappe  falls. 

The  effect  of  the  form  of  the  sill  and  of  the  down-stream 
channel  will  be  considered  later,  but,  for  the  present,  attention 
will  be  confined  to  weirs  with  sharp  edges,  and  to  those  in  which 
the  air  has  free  access  under  the  nappe  so  that  it  detaches  itself 
entirely  from  the  weir  as  shown  in  Fig.  70. 

60.     Rectangular  sharp-edged  weir. 

If  the  crest  and  sides  of  the  weir  are  made  sharp-edged,  as 
shown  in  Fig.  70,  and  the  weir  is  narrower  than  the  approaching 
channel,  and  the  sill  some  distance  above  the  bed  of  the  stream, 
there  is  at  the  sill  and  at  the  sides,  contraction  similar  to  that  at 
a  sharp-edged  orifice. 

The  surface  of  the  water  as  it  approaches  the  weir  falls,  taking 
a  curved  form,  so  that  the  thickness  hs,  Fig.  70,  of  the  vein  over 
the  weir,  is  less  than  H,  the  height,  above  the  sill,  of  the  water  at 
L.  H.  6 


82  HYDRAULICS 

some  distance  from  the  weir.  The  height  H,  which  is  called  the 
head  over  the  weir,  should  be  carefully  measured  at  such  a  distance 
from-  it,  that  the  water  surface  has  not  commenced  to  curve. 
Fteley  -and  Stearns  state,  that  this  distance  should  be  equal  to 
2^  times  the  height  of  the  weir  above  the  bed  of  the  stream. 

For  the  present,  let  it  be  assumed  that  at  the  point  where  H  is 
measured  the  water  is  at  rest.  In  actual  cases  the  water  will 
always  have  some  velocity,  and  the  effect  of  this  velocity  will  have 
to  be  considered  later.  H  may  be  called  the  still  water  head  over 
the  weir,  and  in  all  the  formulae  following  it  has  this  meaning. 

Side  contraction.  According  to  Fteley  and  Stearns  the  amount 
by  which  the  stream  is  contracted  when  the  weir  is  sharp-edged 
is  from  0'06  to  0'12H  at  each  side,  and  Francis  obtained  a  mean  of 
O'lH.  A  wide  weir  may  be  divided  into  several  bays  by  parti- 
tions, and  there  may  then  be  more  than  two  contractions,  at  each 
of  which  the  effective  width  of  the  weir  will  be  diminished,  if 
Francis'  value  be  taken,  by  O'lH. 

If  L  is  the  total  width  of  a  rectangular  weir  and  N  the  number 
of  contractions,  the  effective  width  I,  Fig.  70,  is  then, 

(L-O'INH) 

When  L  is  very  long  the  lateral  contraction  may  be  neglected. 

Suppression  of  the  contraction.  The  side  contraction  can  be 
completely  suppressed  by  making  the  approaching  channel  with 
vertical  sides  and  of  the  same  width  as  the  weir,  as  was  done  for 
the  orifice  shown  in  Fig.  47.  The  width  of  the  stream  is  then 
equal  to  the  width  of  the  sill. 

61.  Derivation  of  the  weir  formula  from  that  of  a  large 
orifice. 

If  in  the  formula  for  large  orifices,  p.  64,  hQ  is  made  equal  to 
zero  and  for  the  effective  width  of  the  stream  the  length  I  is 
substituted  for  6,  and  Jc  is  unity,  the  formula  becomes 

Q=t>/2^.Z.fci*    ........................  (1). 

If  instead  of  hi  the  head  H,  Fig.  70,  is  substituted,  and 
a  coefficient  H  introduced, 


The  actual  width  I  is  retained  instead  of  L,  to  make  allowance 
for  the  end  contraction  which  as  explained  above  is  equal  to  O'lH 
for  each  contraction. 

If  the  width  of  the  approaching  channel  is  made  equal  to  the 
width  of  the  weir  I  is  equal  to  L. 

With  N  contractions     I  =  (L  -  Q'1NJ$ 

and  Q  =  f  C  >/2cj  .  (L  -  O'lN)  H*. 

If  C  is  given  a  mean  value  of  0*625, 

Q  =  3'33  (L-  O'lN)  Hf  .....................  (2). 


FLOW  OVER  WEIRS  83 

This  is  the  well-known  formula  deduced  by  Francis*  from 
a  careful  series  of  experiments  on  sharp-edged  weirs. 

The  formula,  as  an  empirical  one,  is  approximately  correct  and 
gives  reliable  values  for  the  discharge. 

The  method  of  obtaining  it  from  that  for  large  orifices  is, 
however,  open  to  very  serious  objection,  as  the  velocity  at  F  on 
the  section  EF,  Fig.  70,  is  clearly  not  equal  to  zero,  neither  is  the 
direction  of  flow  at  the  surface  perpendicular  to  the  section  EF, 
and  the  pressure  on  EF,  as  will  be  understood  later  (section  83) 
is  not  likely  to  be  constant. 

That  the  directions  and  the  velocities  of  the  stream  lines  are 
different  from  those  through  a  section  taken  near  a  sharp-edged 
orifice  is  seen  by  comparing  the  thickness  of  the  jet  in  the  two 
cases  with  the  coefficient  of  discharge. 

For  the  sharp-edged  orifice  with  side  contractions  suppressed, 
the  ratio  of  the  thickness  of  the  jet  to  the  depth  of  the  orifice  is  not 
very  different  from  the  coefficient  of  discharge,  being  about  0'625, 
but  the  thickness  EF  of  the  nappe  of  the  weir  is  very  nearly  0'78H, 
whereas  the  coefficient  of  discharge  is  practically  0*625,  and  the 
thickness  is  therefore  1'24  times  the  coefficient  of  discharge. 

It  appears  therefore,  that  although  the  assumptions  made  in 
calculating  the  flow  through  an  orifice  may  be  justifiable,  providing 
the  head  above  the  top  of  the  orifice  is  not  very  small,  yet  when 
it  approaches  zero,  the  assumptions  are  not  approximately  true. 

The  angles  which  the  stream  lines  make  with  the  plane  of  EF 
cannot  be  very  different  from  90  degrees,  so  that  it  would  appear, 
that  the  error  principally  arises  from  the  assumption  that  the 
pressure  throughout  the  section  is  uniform. 

Bazin  for  special  cases  has  carefully  measured  the  fall  of  the 
point  F  and  the  thickness  EF,  and  if  the  assumptions  of  constant 
pressure  and  stream  lines  perpendicular  to  EF  are  made,  the 
discharge  through  EF  can  be  calculated. 

For  example,  the  height  of  the  point  E  above  the  sill  of  the 
weir  for  one  of  Bazin's  experiments  was  0'112H  and  the  thickness 
EF  was  0'78H.  The  fall  of  the  point  F  is,  therefore,  O'lOSH. 
Assuming  constant  pressure  in  the  section,  the  discharge  per  foot 
width  of  the  weir  is,  then, 


0108H 

=  §  >/2g  .  H*  {('888)*  -  (-108)*} 


Lowell,  Hydraulic  Experiments,  New  York,  1858. 

6—2 


84  HYDRAULICS 

The  actual  discharge  per  foot  width,  by  experiment,  was 

g  =  0'433N/20.H^, 

so  that  the  calculation  gives  the  discharge  1'228  greater  than  the 
actual,  which  is  approximately  the  ratio  of  the  thickness  EF  to 
the  thickness  of  the  stream  from  a  sharp-edged  orifice  having 
a  depth  H.  The  assumption  of  constant  pressure  is,  therefore, 
quite  erroneous. 

62.     Thomson's  principle  of  similarity. 

"  When  a  frictionless  liquid  flows  out  of  similar  and  similarly 
placed  orifices  in  similar  vessels  in  which  the  same  kind  of  liquid 
is  at  similar  heights,  the  stream  lines  in  the  different  flows  are 
similar  in  form,  the  velocities  at  similar  points  are  proportional  to 
the  square  roots  of  the  linear  dimensions,  and  since  the  areas  of 
the  stream  lines  are  proportional  to  the  squares  of  the  linear 
dimensions,  the  discharges  are  proportional  to  the  linear  dimensions 
raised  to  the  power  of  f  *." 

Let  A  and  B,  Figs.  71  and  72,  be  exactly  similar  vessels  with 
similar  orifices,  and  let  all  the  dimensions  of  A  be  n  times  those 
of  B.  Let  c  and  Ci  be  similarly  situated  areas  on  similar  stream 
lines. 


Fig.  71. 


Fig.  72. 


Then,  since  the  dimensions  of  A  are  n  times  those  of  B,  the 
fall  of  free  level  at  c  is  n  times  that  at  Ci .  Let  v  be  the  velocity 
at  c  and  Vi  at  Ci. 

Then,  since  it  has  been  shown  (page  36)  that  the  velocity  in 
any  stream  line  is  proportional  to  the  square  root  of  the  fall  of 
free  level, 

•'.    v  :  Vi  ::   *Jn  :  1. 

Again  the  area  at  c  is  n2  times  the  area  at  Ci  and,  therefore, 
the  discharge  through  c         o    ,-       5 
the  discharge  through  c,  =  n"  ^n  =  ^^ 
which  proves  the  principle. 

*  British  Association  Eeports  1858  and  1876. 


FLOW   OVER  WEIRS  85 

63.     Discharge     through     a     triangular     notch     by    the 
principle  of  similarity. 

Let  ADC,  Figs.  73  and  74,  be  a  triangular  notch. 
A  C A 

X r -^- — r ^- 


D  D 

Fig.  73.  Fig.  74. 

Let  the  depth  of  the  flow  through  the  notch  at  one  time  be  H 
and  at  another  n .  H. 

Suppose  the  area  of  the  stream  in  the  two  cases  to  be  divided 
into  the  same  number  of  horizontal  elements,  such  as  ab  and  aj>i. 

Then  clearly  the  thickness  of  ab  will  be  n  times  the  thickness 
of  a-J}\. 

Let  Oi&i  be  at  a  distance  x  from  the  apex  B,  and  ab  at  a 
distance  rac;  then  the  width  of  ab  is  clearly  n  times  the  width 
of  Oibi. 

The  area  of  ab  will  therefore  be  ri*  times  the  area  of  a^ . 

Again,  the  head  above  ab  is  n  times  the  head  above  a$i  and 
therefore  the  velocity  through  ab  will  be  \ln  times  the  velocity 
through  aibi  and  the  discharge  through  ab  will  be  r$  times 
that  through  0-161. 

More  generally  Thomson  expresses  this  as  follows : 

"  If  two  triangular  notches,  similar  in  form,  have  water  flowing 
through  them  at  different  depths,  but  with  similar  passages  of 
approach,  the  cross  sections  of  the  jets  at  the  notches  may  be 
similarly  divided  into  the  same  number  of  elements  of  area,  and 
the  areas  of  corresponding  elements  will  be  proportional  to  the 
squares  of  the  lineal  dimensions  of  the  cross  sections,  or  pro- 
portional to  the  squares  of  the  heads." 

As  the  depth  h  of  each  element  can  be  expressed  as  a  fraction 
of  the  head  H,  the  velocities  through  these  elements  are  propor- 
tional to  the  square  root  of  the  head,  and,  therefore,  the  discharge 
is  proportional  to  H*. 

Therefore  Q  oo  H*, 

or  Q  =  C  .  H", 

C  being  a  coefficient  which  has  to  be  determined  by  experiment. 

From  experiments  with  a  sharp-edged  notch  having  an  angle 
at  the  vertex  of  90  degrees,  he  found  C  to  be  practically  constant 
for  all  heads  and  equal  to  2'635.  Then 

a* ..  ...(3). 


HYDRAULICS 

64.     Flow  through  a  triangular  notch. 

The  flow  through  a  triangular  notch  is  frequently  given  as 


in  which  B  is  the  top  width  of  the  notch  and  n  an  experimental  coefficient. 

It  is  deduced  as  follows  : 

Let  ADC,  Fig.  74,  be  the  triangular  notch,  H  being  the  still  water  head  over 
the  apex,  and  B  the  width  at  a  height  H  above  the  apex.     At  any  depth  h  the 

T>    /TT  I-  \ 

width  b  of  the  strip  a-Jb-,  is  -  —  -  -  . 
ti 

If  the  velocity  through  this  strip  is  assumed  to  be  v  =  k*j2gh,  the  width  of  the 

stream  through  a^,  —  —  ,  and  the  thickness  dh,  the  discharge  through  it  is 

H 


The  section  of  the  jet  just  outside  the  orifice  is  really  less  than  the  area  EFD. 
The  width  of  the  stream  through  any  strip  a^  is  less  than  a^,  the  surface  is  lower 
than  EF,  and  the  apex  of  the  jet  is  some  distance  above  B. 

The  diminution  of  the  width  of  a^  has  been  allowed  for  by  the  coefficient  c,  and 
the  diminution  of  depth  might  approximately  be  allowed  for  by  integrating  between 
h  =  0  and  h  =  H,  and  introducing  a  third  coefficient  Cj. 

Then  =  fccc,        B  ^T~  h) 

ti 


Keplacing  cc-Jt  by  n 

Q  =  TVW27.BH£  .......................................  (4). 

Calling  the  angle  ADC,  0, 

B=2Htan|, 


When  B  is  90  degrees,  B  is  equal  to  2H,  and 


Taking  a  mean  value  for  n  of  0'617 
Q  =  2-635.  H*, 

which  agrees  with  Thomson's  formula  for  a  right-angled  notch. 

The  result  is  the  same  as  obtained  by  the  method  of  similarity,  but  the  method 
of  reasoning  is  open  to  very  serious  objection,  as  at  no  section  of  the  jet  are  all  the 
stream  lines  normal  to  the  section,  and  k  cannot  therefore  be  constant.  The 
assumption  that  the  velocity  through  any  strip  is  proportional  to  >Jh  is  also  open 
to  objection,  as  the  pressure  throughout  the  section  can  hardly  be  uniform. 

65.    Discharge     through     a     rectangular    weir    by    the 
principle  of  similarity. 

The  discharge  through  a  rectangular  weir  can  also  be  obtained 
by  the  principle  of  similarity. 


FLOW  OVER   WEIRS 


87 


Consider  two  rectangular  weirs  each  of  length  L,  Figs.  75 
and  76,  and  let  the  head  over  the  sill  be  H  in  the  one  case  and 
Hi,  or  nTL,  in  the  other.  Assume  the  approaching  channel  to  be 
of  such  a  form  that  it  does  not  materially  alter  the  flow  in  either 
case. 


H 


H 


J) 


Fig.  75. 


B 

Fig.  76. 


To  simplify  the  problem  let  the  weirs  be  fitted  with  sides 
projecting  up  stream  so  that  there  is  no  side  contraction. 

Then,  if  each  of  the  weirs  be  divided  into  any  number  of  equal 
parts  the  flow  through  each  of  these  parts  in  any  one  of  the  weirs 
will  be  the  same. 

Suppose  the  first  weir  to  be  divided  into  N  equal  parts.     If 

then,  the  second  weir  is  divided  into       '      equal  parts,  the  parts 

in  the  second  weir  will  be  exactly  similar  to  those  of  the  first. 

By  the  principle  of  similarity,  the  discharge  through  each  of 
the  parts  in  the  first  weir  will  be  to  the  discharge  in  the  second 

as  — j ,  and  the  total  discharge  through  the  first  weir  is  to  the 
HI* 

discharge  through  the  second  as 

N.H^5      _Hi_! 
AT   TT    TT,i      TT,I      *ii' 


H! 

Instead  of  two  separate  weirs  the  two  cases  may  refer  to  the 
same  weir,  and  the  discharge  for  any  head  H  is,  therefore,  pro- 
portional to  H^ ;  and  since  the  flow  is  proportional  to  L 

Q  =  C.L.H*, 

in  which  C  is  a  coefficient  which  should  be  constant. 

66.     Rectangular  weir  with  end  contractions. 

If  the  width  of  the  channel  as  it  approaches  the  weir  is  greater 
than  the  width  of  the  weir,  contraction  takes  place  at  each  side, 
and  the  effectual  width  of  the  stream  or  nappe  is  diminished ;  the 
amount  by  which  the  stream  is  contracted  is  practically  inde- 
pendent of  the  width  and  is  a  constant  fraction  of  H,  as  explained 
above,  or  is  equal  to  &H,  k  being  about  O'l 


88  HYDRAULICS 

Let  the  total  width  of  each  weir  be  now  divided  into  three 
parts,  the  width  of  each  end  part  being  equal  to  n  .  ~k  .  H.  The 
width  of  the  end  parts  of  the  transverse  section  of  the  stream  will 
each  be  (n-  1)  ~k  .  H,  and  the  width  of  central  part  L  -  2nkTL. 

The  flow  through  the  central  part  of  the  weir  will  be  equal  to 


Now,  whatever  the  head  on  the  weir,  the  end  pieces  of  the 
stream,  since  the  width  is  (n  -  1)  &H  and  k  is  a  constant,  will  be 
similar  figures,  and,  therefore,  the  flow  through  them  can  be 
expressed  as 


The  total  flow  is,  therefore, 

Q  =  C  (L  -  2wfcH)  Ht  +  20,  (n  -  1)  fcHH*. 
If  now  Ci  is  assumed  equal  to  C 

Q=C(L-2&H)Hi 

If  instead  of  two  there  are  N  contractions,  due  to  the  weir 
being  divided  into  several  bays  by  posts  or  partitions,  the  formula 
becomes 

Q«0(L-N01.H)'H*. 

This  is  Francis7  formula,  and  by  Thomson's  theory  it  is  thus 
shown  to  be  rational. 

67.    Bazin's*  formula  for  the  discharge  of  a  weir. 

The  discharge  through  a  weir  with  no  side  contraction  may  be 
written 


or 

the  coefficient  m  being  equal  to 

Taking  Francis'  value  for  C  as  3*33,  m  is  then  0'415. 
From  experiments  on  sharp-crested  weirs  with  no  side  con- 
traction Bazin  deduced  for  mt  the  value 

A/mK     '00984 
m  =  0'405  +     TT     . 
±1 

In  Table  IX,  and  Fig.  77,  are  shown  Bazin's  values  for  m  for 
different  heads,  and  also  those  obtained  by  Eafter  at  Cornell  upon 
a  weir  similar  to  that  used  by  Bazin,  the  maximum  head  in  the 
Cornell  experiments  being  much  greater  than  that  in  Bazin's 
experiments.  In  Fig.  77  are  also  shown  several  values  of  m,  as 
calculated  by  the  author,  from  Francis'  experimental  data. 

*  Annales  des  Fonts  et  Chaussees,  1888  —  1898. 

t  "Experiments  on  flow  over  Weirs,"  Am.S.C.E.  Vol.  xxvii. 


FLOW   OVER  WEIRS  89 

TABLE  IX. 

Values  of  the  coefficient  m  in  the  formula  Q  -  mL  \/2gr  H*. 

Weir,  sharp-crested,  6'56  feet  wide  with  free  overfall  and  lateral 
contraction  suppressed,  H  being  the  still  water  head  over  the  weir, 
or  the  measured  head  h*  corrected  for  velocity  of  approach. 

Bazin. 

Head  in  feet    0-164        0*328        O656        0'984        1-312        1-64         1-968 
m  0-448        0-432        0-421        0-417        0-414        0-412       0-409 


m 

11  . 

Rafter. 
Head  in  feet  m  C 

0-1  0-4286  3-437 

0-5  0-4230  3-392 

1-0  0-4174  3-348 

1'5  0-4136  3-317 

2-0  0-4106  3-293 

2-5  0-4094  3-283 

3-0  0-4094  3-283 

3-5  0-4099  3-288 

4-0  0-4112  3-298 

4-5  0-4125  3-308 

5-0  0-4133  3-315 

5-5  0-4135  3-316 

6-0  0-4136  3-317 

68.     Bazin's  and  the  Cornell  experiments  on  weirs. 

Bazin's  experiments  were  made  on  a  weirt  6*56  feet  long 
having  the  approaching  channel  the  same  width  as  the  weir,  so 
that  the  lateral  contractions  were  suppressed,  and  the  discharge 
was  measured  by  noting  the  time  taken  to  fill  a  concrete  trench  of 
known  capacity. 

The  head  over  the  weir  was  measured  by  means  of  the  hook 
gauge,  page  249.  Side  chambers  were  constructed  and  connected 
to  the  channel  by  means  of  circular  pipes  O'l  m.  diameter. 

The  water  in  the  chambers  was  very  steady,  and  its  level 
could  therefore  be  accurately  gauged.  The  gauges  were  placed 
5  metres  from  the  weir.  The  maximum  head  over  the  weir  in 
Bazin's  experiments  was  however  only  2  feet. 

The  experiments  for  higher  heads  at  Cornell  University  were 
made  on  a  weir  of  practically  the  same  width  as  Bazin's,  6'53  feet, 
the  other  conditions  being  made  as  nearly  the  same  as  possible  ; 
the  maximum  head  on  the  weir  was  6  feet. 

*  See  page  90. 

t  Annales  des  Fonts  et  Chaussees,  p.  445,  Vol.  n.  1891. 


90 


HYDRAULICS 


The  results  of  these  experiments,  Fig.  77,  show  that  the 
coefficient  m  diminishes  and  then  increases,  having  a  minimum 
value  when  H  is  between  2'5  feet  and  3  feet. 


44 


-43 


V3 


Head  in,  Feet. 

Mean,  coefficient,  carves  for  Sharp -edged.  Weirs 
*!•  BOL^LTLS  E.ccperi?Tiefit$ 

n       (Deduced,  by  the  author) 

Fig.  77. 


o  CorneLL 
A  Fronds 


It  is  doubtful,  however,  although  the  experiments  were  made 
with  great  care  and  skill,  whether  at  high  heads  the  deduced 
coefficients  are  absolutely  reliable. 

To  measure  the  head  over  the  weir  a  1  inch  galvanised  pipe 
with  holes  Jinch  diameter  and  opening  downwards,  6  inches 
apart,  was  laid  across  the  channel.  To  this  pipe  were  connected 
f  inch  pipes  passing  through  the  weir  to  a  convenient  point  below 
the  weir  where  they  could  be  connected  to  the  gauges  by  rubber 
tubing.  The  gauges  were  glass  tubes  f  inch  diameter  mounted 
on  a  frame,  the  height  of  the  water  being  read  on  a  scale 
graduated  to  2mm.  spaces. 

69.     Velocity  of  approach. 

It  should  be  clearly  understood  that  in  the  formula  given,  it 
has  been  assumed,  in  giving  values  to  the  coefficient  m  that  H  is 
the  height,  above  the  sill  of  the  weir,  of  the  still  water  surface. 


FLOW  OVER   WEIRS  91 

In  actual  cases  the  water  where  the  head  is  measured  will  have 
some  velocity,  and  due  to  this,  the  discharge  over  the  weir  will  be 
increased. 

If  Q  is  the  actual  discharge  over  a  weir,  and  A  is  the  area  of 
the  up-stream  channel  approaching  the  weir,  the  mean  velocity  in 

the  channel  is  v  =  -^  . 
A 

There  have  been  a  number  of  methods  suggested  to  take  into 
account  this  velocity  of  approach,  the  best  perhaps  being  that 
adopted  by  Hamilton  Smith,  and  Bazin. 

This  consists  in  considering  the  equivalent  still  water  head  H, 
over  the  weir,  as  equal  to 


a  being  a   coefficient    determined    by   experiment,    and    h   the 
measured  head. 

The  discharge  is  then 


(5), 


2 

Expanding  (5),  and  remembering  that  ~— r  is  generally  a  small 


quantity, 


The  velocity  v  depends  upon  the  discharge  Q  to  be  determined 
and  is  equal  to  -    . 


Therefore         Q  =  mL  hj2gh  ( 


1  + 


From  five  sets  of  experiments,  the  height  of  the  weir  above  the 
bottom  of  the  channel  being  different  for  each  set,  Bazin  found 
the  mean  value  of  a  to  be  1*66. 

This  form  of  the  formula,  however,  is  not  convenient  for  use, 
since  the  unknown  Q  appears  upon  both  sides  of  the  equation. 

If,  however,  the  discharge  Q  is  expressed  as 

Q  -  nL  ^2gh.  h, 
the  coefficient  n  for  any  weir  can  be  found  by  measuring  Q  and  h. 

It  will  clearly  be  different  from  the  coefficient  m,  since  for  m 
to  be  used  h  has  to  be  corrected. 

From  his  experimental  results  Bazin  calculated  n  for  various 
heads,  some  of  which  are  shown  in  Table  X. 


92  HYDRAULICS 

Substituting  this  value  of  Q  in  the  above  formula, 

3  a  - 


/(_, 
(7). 


Let  fan2  be  called  fe. 
Then  Q  - 


1  + 


Or,  when  the  width  of  channel  of  approach  is  equal  to  "the 
width  of  the  weir,  and  the  height  of  the  sill,  Fig.  78,  is  p  feet  above 
the  bed  of  the  channel,  and  h  the  measured  head, 


and 


Fig.  78. 

The  mean  value  given  to  the  coefficient  Jc  by  Bazin  is  0'55, 
so  that 


This  may  be  written 


(9). 


Q  = 


\r2gh, 


in  which 


Substituting  for  m  the  value  given  on  page  88, 

\ 


mi  may  be  called  the  absolute  coefficient  of  discharge. 

The  coefficient  given  in  the  Tables. 

It  should  be  clearly  understood  that  in  determining  the  values 
of  m  as  given  in  the  Tables  and  in  Fig.  77  the  measured  head  h 
was  corrected  for  velocity  of  approach,  and  in  using  this 


FLOW  OVER  WEIRS  93 

coefficient    to    determine    Q,   h    must    first   be   corrected,   or   Q 
calculated  from  formula  9. 

Rafter  in  determining  the  values  of  ra  from  the  Cornell  ex- 


periments, increased  the  observed  head  h  by  ~  only,  instead  of 
by  1-66  g. 

Fteley  and  Stearns*,  from  their  researches  on  the  flow  over 
weirs,  found  the  correction  necessary  for  velocity  of  approach  to 
be  from 

1-45  to  1'5  £. 
20 

Hamilton  Smith  t  adopts  for  weirs  with  end  contractions 
suppressed  the  values 

T33  to  1-40 1^, 

and  for  a  weir  with  two  end  contractions, 
11  to  1-25  |^. 

TABLE  X. 
Coefficients  n  and  m  as  calculated  by  Bazin  from  the  formulae 


and  Q=mW20H*, 

h  being  the  head  actually  measured  and  H  the  head  corrected  for 

velocity  of  approach. 

Coefficient 

0-448 
0-417 
0-4118 

An  example  is  now  taken  illustrating  the  method  of  deducing 
the  coefficients  n  and  m  from  the  result  of  an  experiment,  and  the 
difference  between  them  for  a  special  case. 

Example.  In  one  of  Bazin's  experiments  the  width  of  the  weir  and  the 
approaching  channel  were  both  6 -56  feet.  The  depth  of  the  channel  approaching 
the  weir  measured  at  a  point  2  metres  up  stream  from  the  weir  was  7'544  feet  and 
the  head  measured  over  the  weir,  which  may  be  denoted  by  h,  was  0*984  feet.  The 
measured  discharge  was  21-8  cubic  ft.  per  second. 

*  Transactions  Am.S.C.E.,  Vol.  xn. 
f  Hydraulics. 


Head 
h  in  feet 

Height  of  sill 
p  in  feet 

Coefficient 
n 

0-164 

0-656 

0-458 

6-560 

0-448 

0-984 

0-656 

0-500 

6-560 

0-421 

1-640 

0-656 

0-500 

6-560 

0-421 

94  HYDRAULICS 

The  velocity  at  the  section  where  h  was  measured,  and  which  may  be  called  the 
velocity  of  approach  was,  therefore, 


7-544x6-56'     7-544x6-56 
=  0-44  feet  per  second. 
If  now  the  formula  for  discharge  be  written 


and  n  is  calculated   from   this  formula    by   substituting   the    known   values  of 
Q,  L  and  h 

7i=0-421. 
Correcting  h  for  velocity  of  approach, 


Then 
from  which 


.  -9888 


It  will  seem  from  Table  X  that  when  the  height  p  of  the  sill  of  the  weir  above 
the  stream  bed  is  small  compared  with  the  head,  the  difference  may  be  much 
larger  than  for  this  example. 

When  the  head  is  1-64  feet  and  larger  than  p,  the  coefficient  n  is  eighteen 
per  cent,  greater  than  m.  In  such  cases  failure  to  correct  the  coefficient  will  lead 
to  considerable  inaccuracy. 

70.  Influence  of  the  height  of  the  weir  sill  above  the  bed 
of  the  stream  on  the  contraction. 

The  nearer  the  sill  is  to  the  bottom  of  the  stream,  the  less  the 
contraction  at  the  sill,  and  if  the  depth  is  small  compared  with  H, 
the  diminution  on  the  contraction  may  considerably  affect  the 
flow. 

When  the  sill  was  1*15  feet  above  the  bottom  of  a  channel, 

of  the  same  width  as  the  weir,  Bazin  found  the  ratio  ^   (Fig.  85) 

JH 

to  be  0*097,  and  when  it  was  3*70  feet,  to  be  0112.     For  greater 

p 

heights  than  these  the  mean  value  of  ^  was  0'13. 

±1 

71.  Discharge   of  a   weir    when   the   air    is   not   freely 
admitted  beneath  the  nappe.     Form  of  the  nappe. 

Francis  in  the  Lowell  experiments,  found  that,  by  making  the 
width  of  the  channel  below  the  weir  equal  to  the  width  of  the 
weir,  and  thus  preventing  free  access  of  air  to  the  underside  of  the 
nappe,  the  discharge  was  increased.  Bazin*,  in  the  experiments 
already  referred  to,  has  investigated  very  fully  the  effect  upon 
the  discharge  and  upon  the  form  of  the  nappe,  of  restricting  the 
free  passage  of  the  air  below  the  nappe.  He  finds,  that  when  the 
flow  is  sufficient  to  prevent  the  air  getting  under  the  nappe,  it  may 
assume  one  of  three  distinct  forms,  and  that  the  discharge  for 
*  Annales  des  Fonts  et  Chaussees,  1891  and  1898. 


FLOW   OVER   WEIRS 


95 


one  of  them  may  be  28  per  cent,  greater  than  when  the  air  is 
freely  admitted,  or  the  nappe  is  "free."  Which  of  these  three 
forms  the  nappe  assumes  and  the  amount  by  which  the  discharge 
is  greater  than  for  the  "free  nappe,"  depends  largely  upon  the 
head  over  the  weir,  and  also  upon  the  height  of  the  weir  above 
the  water  in  the  down-stream  channel. 

The  phenomenon  is,  however,  very  complex,  the  form  of  the 
nappe  for  any  head  depending  to  a  very  large  extent  upon 
whether  the  head  has  been  decreasing,  or  increasing,  and  for  a 
given  head  may  possibly  have  any  one  of  the  three  forms,  so  that 
the  discharge  is  very  uncertain.  M.  Bazin  distinguishes  the  forms 
of  nappe  as  follows  : 

(1)  Free  nappe.     Air  under  nappe  at  atmospheric  pressure, 
Figs.  70  and  78. 

(2)  Depressed  nappe  enclosing  a  limited  volume  of  air  at  a 
pressure  less  than  that  of  the  atmosphere,  Fig.  79. 

(3)  Adhering  nappe.     No  air  enclosed  and  the  nappe  adher- 
ing to  the  down-stream  face  of  the  weir,  Fig.  80.    The  nappe  in  this 
case  may  take  any  one  of  several  forms. 


Top  of  (haravd\ 


Fig.  80. 

No  air  enclosed  but 
the  nappe  encloses  a  mass  of  turbulent  water  which  does  not  move 
with  the  nappe,  and  which  is  said  to  wet  the  nappe. 


Fig.  79. 
(4)     Drowned  or  wetted  nappe,  Fig.  81. 


Fig.  81. 


96  HYDRAULICS 

72.  Depressed  nappe. 

The  air  below  the  nappe  being  at  less  than  the  atmospheric 
pressure  the  excess  pressure  on  the  top  of  the  nappe  causes  it  to 
be  depressed.  There  is  also  a  rise  of  water  in  the  down-stream 
channel  under  the  nappe. 

The  discharge  is  slightly  greater  than  for  a  free  nappe.  On  a 
weir  2*46  feet  above  the  bottom  of  the  up-stream'  channel,  the 
nappe  was  depressed  for  heads  below  0*77  feet,  and  at  this  head 
the  coefficient  of  discharge  was  1'08  mly  ml  being  the  absolute 
coefficient  for  the  free  nappe. 

73.  Adhering  nappes. 

As  the  head  for  this  weir  approached  0*77  feet  the  air  was 
rapidly  expelled,  and  the  nappe  became  vertical  as  in  Fig.  80,  its 
surface  having  a  corrugated  appearance.  The  coefficient  of  dis- 
charge changed  from  1*08  m:  to  1*28  m^  This  large  change  in 
the  coefficient  of  discharge  caused  the  head  over  the  weir  to  fall 
to  0'69  feet,  but  the  nappe  still  adhered  to  the  weir. 

74.  Drowned  or  wetted  nappes. 

As  the  head  was  further  increased,  and  approached  0'97  feet, 
the  nappe  came  away  from  the  weir  face,  assuming  the  drowned 
form,  and  the  coefficient  suddenly  fell  to  1*19  mi.  As  the  head 
was  further  increased  the  coefficient  diminished,  becoming  1'12 
when  the  head  was  above  1'3  feet. 

The  drowned  nappes  are  more  stable  than  the  other  two,  but 
whereas  for  the  depressed  and  adhering  nappes  the  discharge  is 
not  affected  by  the  depth  of  water  in  the  down-stream  channel, 
the  height  of  the  water  may  influence  the  flow  of  the  drowned 
nappe.  If  when  the  drowned  nappe  falls  into  the  down  stream 
the  rise  of  the  water  takes  place  at  a  distance  from  the  foot  of  the 
nappe,  Fig.  81,  the  height  of  the  down-stream  water  does  not  affect 
the  flow.  On  the  other  hand  if  the  rise  encloses  the  foot  of  the 
nappe,  Fig.  82,  the  discharge  is  affected.  Let  h2  be  the  difference 


Fig.  82. 


FLOW   OVER  WEIRS  97 

of  level  of  the  sill  of  the  weir  and  the  water  below  the  weir.  The 
coefficient  of  discharge  in  the  first  case  is  independent  of  /i2  but  is 
dependent  upon  p  the  height  of  the  sill  above  the  head  of  the  up- 
stream channel,  and  is 


...(11). 

Bazin  found  that  the  drowned  nappe  could  not  be  formed  if  h 
is  less  than  0*4  p  and,  therefore,  j?  cannot  be  greater  than  2*5. 
Substituting  for  ma  its  value 


from  (10)  page  92 

mo  =  0-470  +  0-0075 1*  (12). 

In  the  second  case  the  coefficient  depends  upon  h^,  and  is, 

mo  =  wi(l'06  +  0-16>)('--0-05>)?   .  ...(13), 

/  \p  /  h 

for  which,  with   a   sufficient   degree   of   approximation,  may  be 
substituted  the  simpler  formula, 

7710^/1-05  + 1-15^  ...(14). 


The  limiting  value  of  m0  is  l'2wi,  for  if  /^  becomes  greater 
than  h  the  nappe  is  no  longer  drowned. 

Further,  the  rise  can  only  enclose  the  foot  of  the  nappe  when 
h2  is  less  than  (f  p  -  h} .  As  h2  passes  this  value  the  rise  is  pushed 
down  stream  away  from  the  foot  of  the  nappe  and  the  coefficient 
changes  to  that  of  the  preceding  case. 

75.     Instability  of  the  form  of  the  nappe. 

The  head  at  which  the  form  of  nappe  changes  depends  upon 
whether  the  head  is  increasing  or  diminishing,  and  the  depressed 
and  adhering  nappes  are  very  unstable,  an  accidental  admission 
of  air  or  other  interference  causing  rapid  change  in  their  form. 
Further,  the  adhering  nappe  is  only  formed  under  special  circum- 
stances, and  as  the  air  is  expelled  the  depressed  nappe  generally 
passes  directly  to  the  drowned  form. 

If,  therefore,  the  air  is  not  freely  admitted  below  the  nappe 
the  form  for  any  given  head  is  very  uncertain  and  the  discharge 
cannot  be  obtained  with  any  great  degree  of  assurance. 

With  the  weir  2*46  feet  above  the  bed  of  the  channel  and  6*56 
feet  long  Bazin  obtained  for  the  same  head  of  0*656  feet,  the  four 
kinds  of  nappe,  the  coefficients  of  discharge  being  as  follows : 
L.  H.  7 


98  HYDRAULICS 

Free  nappe,  0*433 

Depressed  nappe,  0*460 
Drowned  nappe,   level   of  water  down  stream 

0*41  feet  below  the  crest  of  the  weir,  0*497 

Nappe  adhering  to  down-stream  face,  0*554 
The  discharge  for  this  weir  while  the  head  was  kept  constant, 
thus  varied  26  per  cent. 

76.     Drowned  weirs  with  sharp  crests*. 

When  the  surface  of  the  water  down  stream  is  higher  than  the 
sill  of  the  weir,  as  in  Fig.  83,  the  weir  is  said  to  be  drowned. 


Fig.  83. 

Bazin  gives  a  formula  for  deducing  the  coefficients  for  such  a 
weir  from  those  for  the  sharp-edged  weirs  with  a  free  nappe,  which 
in  its  simplest  form  is, 

.-.(15), 


hz  being  the  height  of  the  down-stream  water  above  the  sill  of 
the  weir,  h  the  head  actually  measured  above  the  weir,  p  the 
height  of  the  sill  above  the  up-stream  channel,  and  rax  the 
coefficient  ((10),  p.  92)  for  a  sharp-edged  weir.  This  expression 
gives  the  same  value  within  1  or  2  per  cent,  as  the  formulae  (13) 
and  (14). 

Example.  The  head  over  a  weir  is  1  foot,  and  the  height  of  the  sill  above  the 
up-stream  channel  is  5  feet.  The  length  is  8  feet  and  the  surface  of  the  water 
in  the  down-stream  channel  is  6  inches  above  the  sill.  Find  the  discharge. 

From  formula  (10),  page  92,  the  coefficient  mx  for  a  sharp-edged  weir  with  free 
nappe  is 


=  •4215. 

*  Attempts  have  been  made  to  express  the  discharge  over  a  drowned  weir  as 
equivalent  to  that  through  a  drowned  orifice  of  an  area  equal  to  Lft2,  under  a  head 
h  -  7i2»  together  with  a  discharge  over  a  weir  of  length  L  when  the  head  ish-h^. 

The  discharge  is  then 


n  and  m  being  coefficients. 


FLOW  OVER  WEIRS  99 


Therefore  m0  =  -4215  [1  -05  (1  +  -021)  0-761] 

=  •3440. 


Then  Q  =  -344  x 

=  22-08  cubic  ft.  per  second. 

77.     Vertical  weirs  of  small  thickness. 

Instead  of  making  the  sill  of  a  weir  sharp-edged,  it  may 
have  a  flat  sill  of  thickness  c.  This  will  frequently  be  the  case  in 
practice,  the  weir  being  constructed  of  timbers  of  uniform  width 
placed  one  upon  the  other.  The  conditions  of  flow  for  these  weirs 
may  be  very  different  from  those  of  a  sharp-edged  weir. 

The  nappes  of  such  weirs  present  two  distinct  forms,  according 
as  the  water  is  in  contact  with  the  crest  of  the  weir,  or  becomes 
detached  at  the  up-stream  edge  and  leaps  over  the  crest  without 
touching  the  down-stream  edge.  In  the  second  case  the  discharge 
is  the  same  as  if  the  weir  were  sharp-edged.  When  the  head  h 
over  the  weir  is  more  than  2c  this  condition  is  realised,  and  may 
obtain  when  h  passes  f  c.  Between  these  two  values  the  nappe  is 
in  a  condition  of  unstable  equilibrium  ;  when  h  is  less  than  f  c  the 
nappe  adheres  to  the  sill,  and  the  coefficient  of  discharge  is 


m,  (0'70  +  0185^), 


any  external  perturbation  such  as  the  entrance  of  air  or  the 
passage  of  a  floating  body  causing  the  detachment. 

If  the  nappe  adheres  between  f  c  and  2c  the  coefficient  m0  varies 
from  '98wi  to  l'07mi,  but  if  it  is  free  the  coefficient  m0  =  mi. 
When  H  =  |c,  m0  is  *79rai.  If  therefore  the  coefficients  for  a 
sharp-edged  weir  are  used  it  is  clear  the  error  may  be  con- 
siderable. 

The  formula  for  ra0  gives  approximately  correct  results  when 
the  width  of  the  sill  is  great,  from  3  to  7  feet  for  example. 

If  the  up-stream  edge  of  the  weir  is  rounded  the  discharge  is 
increased.  The  discharge*  for  a  weir  having  a  crest  6*56  feet 
wide,  when  the  up-stream  edge  was  rounded  to  a  radius  of  4  inches, 
was  increased  by  14  per  cent.,  and  that  of  a  weir  2*624  feet  wide 
by  12  per  cent. 

The  rounding  of  the  corners,  due  to  wear,  of  timber  weirs  of 
ordinary  dimensions,  to  a  radius  of  1  inch  or  less,  will,  therefore, 
affect  the  flow  considerably. 

78.     Depressed  and  wetted  nappes  for  flat-crested  weirs. 

The  nappes  of  weirs  having  flat  sills  may  be  depressed,  and 
may  become  drowned  as  for  sharp-edged  weirs. 

*  Annales  des  Fonts  et  Chaiisstes,  Vol.  n.  1896. 

7—2 


100 


HYDRAULICS 


The  coefficient  of  discharge  for  the  depressed  nappes,  whether 
the  nappe  leaps  over  the  crest  or  adheres  to  it,  is  practically  the 
same  as  for  the  free  nappes,  being  slightly  less  for  low  heads  and 
becomes  greater  as  the  head  increases.  In  this  respect  they  differ 
from  the  sharp-crested  weirs,  the  coefficients  for  which  are  always 
greater  for  the  depressed  nappes  than  for  the  free  nappes. 

79.    Drowned  nappes  for  flat-crested  weirs. 

As  long  as  the  nappe  adheres  to  the  sill  the  coefficient  m  may 
be  taken  the  same  as  when  the  nappe  is  free,  or 

'OTO+^. 


When  the  nappe  is  free  from  the  sill  and  becomes  drowned, 
the  same  formula 

m0  =  w1(0'878+  0128 1)  , 

as  for  sharp-crested  weirs  with  drowned  nappes,  may  be  used. 
For  a  given  limiting  value  of  the  head  h  these  two  formulae  give 
the  same  value  of  m0.  When  the  head  is  less  than  this  limiting 
value,  the  former  formula  should  be  used.  It  gives  values  of  m 
slightly  too  small,  but  the  error  is  never  more  than  3  to  4  per  cent. 
When  the  head  is  greater  than  the  limiting  value,  the  second 
formula  should  be  used.  The  error  in  this  case  may  be  as 
great  as  8  per  cent. 

80.    Wide  flat-crested  weirs. 

When  the  sill  is  very  wide  the  surface  of  the  water  falls 
towards  the  weir,  but  the  stream  lines,  as  they  pass  over  the  weir, 
are  practically  parallel  to  the  top  of  the  weir. 

Let  H  be  the  height  of  the  still  water  surface,  and  h  the  depth 
of  the  water  over  the  weir,  Fig.  84. 


Fig.  84. 

Then,  assuming  that  the  pressure  throughout  the  section  of  the 
nappe  is  atmospheric,  the  velocity  of  any  stream  line  is 


(16). 


and  if  L  is  the  length  of  the  weir,  the  discharge  is 


FLOW   OVER   WEIRS  101 

For  the  flow  to  be  permanent  (see  page  106)  Q  must  be  a 
maximum  for  a  given  value  of  h,  or  -^  must  equal  zero. 
Therefore 


From  which  2  (H  -  h)  -  h  =  0, 

and  h  =  f  H. 

Substituting  for  h  in  (16) 


-  0-385L  \20H  .  H  =  3'08L  VS  .  H. 

The  actual  discharge  will  be  a  little  less  than  this  due  to 
friction  on  the  sill,  etc. 

Bazin  found  for  a  flat-crested  weir  6'56  feet  wide  the  coefficient 
ra  was  G'373,  or  C  =2'991. 

Lesbros'  experiments  on  weirs  sufficiently  wide  to  approximate 
to  the  conditions  assumed,  gave  *35  for  the  value  of  the  co- 
efficient m. 

In  Table  XI  the  coefficient  C  for  such  weirs  varies  from  2'66 
to  3-10. 

81.     Flow  over  dams. 

Weirs  of  various  forms.  M.  Bazin  has  experimentally  investi- 
gated the  flow  over  weirs  having  (a)  sharp  crests  and  (b)  flat 
crests,  the  up-  and  down-stream  faces,  instead  of  both  being  vertical, 
being 

(1)  vertical  on  the  down-stream  face  and  inclined  on  the 
up-stream  face, 

(2)  vertical  on  the  up-stream  face  and  inclined  on  the  down- 
stream face, 

(3)  inclined  on  both  the  up-  and  down-stream  faces, 
and  (c)  weirs  of  special  sections. 

The  coefficients  vary  very  considerably  from  those  for  sharp- 
crested  vertical  weirs,  and  also  for  the  various  kinds  of  weirs. 
Coefficients  are  given  in  Table  XI  for  a  few  cases,  to  show  the 
necessity  of  the  care  to  be  exercised  in  choosing  the  coefficient  for 
any  weir,  and  the  errors  that  may  ensue  by  careless  evaluation  of 
the  coefficient  of  discharge. 

For  a  full  account  of  these  experiments  and  the  coefficients 
obtained,  the  reader  is  referred  to  Bazin's*  original  papers,  or  to 
Rafter's!  paper,  in  which  also  will  be  found  the  results  of  experi- 

*  Annale*  des  Fonts  et  Cliaux»€es,  1898. 

t  Transactions  of  tlic  Am.S.C.E.,  Vol.  XLIV.,  1900. 


102 


HYDRAULICS 


TABLE  XL 

Values  of  the  coefficient  C  in  the  formula  Q  =  CL  .  h*,  for  weirs 
of  the  sections  shown,  for  various  values  of  the  observed  head  h. 

Bazin. 


Section  of 
weir 


Head  in  feet 


0-3        0-5        1-0        1-3        2-0        3-0        4-0        5-0        6-0 


1-31$ 


2-66 


2-66 


2-90 


3-10 


3-61 


3-80 


4-01 


3-91 


4-02 


4-15 


4-18 


4-15 


3-46 


3-57 


3-86 


3-80 


3-46 


3-08 


3-49 


3-08 


3-59 


3-19 


3-63 


3-22 


FLOW   OVER  WEIRS 


103 


TABLE  XI   (continued). 
Bazin. 


Section  of 
weir 


-66 


Head  in  feet 


3-10 


2-75 


0-5 


3-27 


3-05 


1-0        1-3 


3-73 


3-52 


3-90 


3-73 


2-0        3-0        4-0        5-0        6-0 


Rafter. 


Section  of 
weir 


Head  in  feet 


0-3        0-5        1-0        1-3        2-0        3-0        4-0        5-0       6-0 


3-35 


3-14 


2-95 


3-68 


3-42 


3-16 


3-83 


3-52 


3-27 


3-77 


3-61 


3-45 


3-68 


3-66 


3'56 


3-70 


3'66 


3-61 


3-71 


3-64 


3-65 


3-71 


3-63 


3-67 


104  HYDRAULICS 

ments  made  at  Cornell  University  on  the  discharge  of  weirs,  similar 
to  those  used  by  Bazin  and  for  heads  higher  than  he  used,  and 
also  weirs  of  sections  approximating  more  closely  to  those  of 
existing  masonry  dams,  used  as  weirs.  From  Bazin's  and  Rafter's 
experiments,  curves  of  discharge  for  varying  heads  for  some  of 
these  actual  weirs  have  been  drawn  up. 

82.  Form  of  weir  for  accurate  gauging. 

The  uncertainty  attaching  itself  to  the  correction  to  be  applied 
to  the  measured  head  for  velocity  of  approach,  and  the  difficulty 
of  making  proper  allowance  for  the  imperfect  contraction  at  the 
sides  and  at  the  sill,  when  the  sill  is  near  the  bed  of  the  channel 
and  is  not  sharp-edged,  and  the  instability  of  the  nappe  and 
uncertainty  of  the  form  for  any  given  head  when  the  admission  of 
air  below  the  nappe  is  imperfect,  make  it  desirable  that  as  far  as 
possible,  when  accurate  gaugings  are  required,  the  weir  should 
comply  with  the  following  four  conditions,  as  laid  down  by 
Bazin. 

(1)  The  sill  of  the  weir  must  be  made  as  high  as  possible 
above  the  bed  of  the  stream. 

(2)  Unless  the  weir  is  long  compared  with  the  head,  the 
lateral  contraction  should  be  suppressed  by  making  the  channel 
approaching  the  weir  with  vertical  sides  and  of  the  same  width  as 
the  weir. 

(3)  The  sill  of  the  weir  must  be  made  sharp-crested. 

(4)  Free  access  of  air  to  the  sides  and  under  the  nappe  of 
the  weir  must  be  ensured. 

83.  Boussinesq's  *  theory  of  the  discharge  over  a  weir. 

As  stated  above,  if  air  is  freely  admitted  below  the  nappe  of 
a  weir  there  is  a  contraction  of  the  stream  at  the  sharp  edge  of  the 
sill,  and  also  due  to  the  falling  curved  surface. 

If  the  top  of  the  sill  is  well  removed  from  the  bottom  of  the 
channel,  the  amount  by  which  the  arched  under  side  of  the  nappe 
is  raised  above  the  sill  of  the  weir  is  assumed  by  Boussinesq — and 
this  assumption  has  been  verified  by  Bazin's  experiments — to  be 
some  fraction  of  the  head  H  on  the  weir. 

Let  CD,  Fig.  85,  be  the  section  of  the  vein  at  which  the 
maximum  rise  of  the  bottom  of  the  vein  occurs  above  the  sill,  and 
let  e  be  the  height  of  D  above  S. 

Let  it  be  assumed  that  through  the  section  CD  the  stream 
lines  are  moving  in  curved  paths  normal  to  the  section,  and  that 
they  have  a  common  centre  of  curvature  0. 

*  Cowptes  Rendus,  1887  and  1889. 


FLOW   OVER   WEIRS 


105 


Let  H  be  the  height  of  the  surface  of  the  water  up  stream 
above  the  sill.  Let  R  be  the  radius  of  the  stream  line  at  any 
point  E  in  CD  at  a  height  x  above  S,  and  RI  and  R2  the  radii  of 
curvature  at  D  and  C  respectively.  Let  Y,  Vi  and  V2  be  the 
velocities  at  E,  D,  and  C  respectively. 


0 


Fig.  85. 


Consider  the  equilibrium  of  any  element  of  fluid  at  the  point 
E,  the  thickness  of  which  is  SR  and  the  horizontal  area  is  a.  If  w 
is  the  weight  of  unit  volume,  the  weight  of  the  element  is  w .  a8R. 

Since  the  element  is  moving  in  a  circle  of  radius  R  the  centri- 


fugal force  acting  on  the  element  is  wa 


Y28R 


Ibs. 


The  force  acting  on  the  element  due  to  gravity  is  wa  SR  Ibs. 
Let  p  be  the  pressure  per  unit  area  on  the  lower  face  of  the 
element  and  p  +  &p  on  the  upper  face. 

Then,  equating  the  upward  and  downward  forces, 

(p  +  ftp)  a  +  waSR  =  pa  + 


Fromwhioh  -  §§=-!  +  -%  (D- 

w  cZR  #R 

Assuming  now  that  Bernoulli's  theorem  is  applicable  to  the 
stream  line  at  EF, 

+  p  +V!  =  H 

w      2g 
Differentiating,  and  remembering  H  is  constant, 

=  0, 


9 


or 


w  dx 


g  .dx  ' 


106       *  HYDRAULICS 

dp      dp 
And  since  fo  =  dR> 

V2        VdV 
tnereiore  -^-  =    •    ,T>    > 

or  RdV  +  VdR  =  0. 

Integrating,  VR  =  constant. 

Therefore  VR  -  VA  =  V2R2 . 

At  the  upper  and  lower  surfaces  of  the  vein  the  pressure  is 
atmospheric,  and  therefore, 


Since  VR  =  ViRi,  and  R  from  the  figure  is  (Ra  +  x  —  e),  therefore, 


The  total  flow  over  the  weir  is 


(3). 


Now  if  the  flow  over  the  weir  is  permanent,  the  thickness  h0  of 
the  nappe  must  adjust  itself,  so  that  for  the  given  head  H  the 
discharge  is  the  maximum  possible. 

The  maximum  flow  however  can  only  take  place  if  each 
filament  at  the  section  GrF  has  the  maximum  velocity  possible  to 
the  conditions,  otherwise  the  filaments  will  be  accelerated  ;  and 
for  a  given  discharge  the  thickness  h0  is  therefore  a  minimum,  or 
for  a  given  value  of  h0  the  discharge  is  a  maximum.  That  is,  when 

Q  is  a  maximum,  -55*  =  0. 
(trio 

If  therefore  RI  can  be  written  as  a  function  of  hQ)  the  value  of 
ho,  which  makes  Q  a  maximum,  can  be  determined  by  differ- 

entiating (3)  and  equating   -~  to  zero. 


Then,  since  Ra  =  RI  +  ho, 

V22     H  -  e  -  i 
and  „-____ 


FLOW   OVER   WEIRS  107 

Therefore,  h0  =  (H  -  e)  (1  -  ™2), 

and  KI  =  n  (1  +  n)  (H  —  e). 

Substituting  this  value  of  Ba  in  the  expression  for  Q, 


which,  since  Q  is  a  maximum  when  -^  =  0,  and  h  is  a  function 

Ctrl 

of  n.  is  a  maximum  when  -~  =  0. 

dn 

Differentiating  and  equating  to  zero, 


the  solution  of  which  gives 

n  =  G'4685, 

and  therefore,  Q  =  0'5216  *J%g  (H  -  e) 

=  0-5216  J2g(l-  4 

=  0'5216  (l  -  4: 


the  coefficient  m  being  equal  to 

0*5216  (l  -  g 
M.  Bazin  has  found  by  actual  measurement,  that  the  mean 

p 

value  for  TT,   when   the  height  of   the  weir  is  at  considerable 
distance  from  the  bottom  of  the  channel,  is  0*13. 
Then,  (l-^=  Q-812, 

and  m  =  0*423. 

It  will  be  seen  on  reference  to  Fig.  77,  that  this  value  is  very 
near  to  the  mean  value  of  m  as  given  by  Francis  and  Bazin,  and 
the  Cornell  experiments.  Giving  to  g  the  value  32'2, 

Q  =  3'39  H^  per  foot  length  of  the  weir. 

If  the  length  of  the  weir  is  L  feet  and  there  are  no  end  con- 
tractions the  total  discharge  is 


and  if  there  are  N  contractions 

Q  =  3-39(L-NO'lH)Hi 


108  HYDRAULICS 

The  coefficient  3'39  agrees  remarkably  well  with  the  mean 
value  of  C  obtained  from  experiment. 

The  value  of  a  theory  must  be  measured  by  the  closeness  of 
the  results  of  experience  with  those  given  by  the  theory,  and  in 
this  respect  Boussinesq's  theory  is  the  most  satisfactory,  as  it  not 
only,  in  common  with  the  other  theories,  shows  that  the  flow  is 
proportional  to  H^,  but  also  determines  the  value  of  the 
constant  C. 

84.  Solving  for  Q,  by  approximation,  when  the  velocity 
of  approach  is  unknown. 

A  simple  method  of  determining  the  discharge  over  a  weir 
when  the  velocity  of  approach  is  unknown,  is,  by  approximation, 
as  follows. 

Let  A  be  the  cross-sectional  area  of  the  channel. 

First  find  an  approximation  to  Q,  without  correcting  for 
velocity  of  approach,  from  the  formula 

Q  =  mLh  */2gh. 
The  approximate  velocity  of  approach  is,  then, 


and  H  is  approximately 


A  nearer  approximation  to  Q  can  then  be  obtained  by  sub- 
stituting H  for  h,  and  if  necessary  a  second  value  for  v  can  be 
found  and  a  still  nearer  approximation  to  H. 

In  practical  problems  this  is,  however,  hardly  necessary. 

Example.  A  weir  without  end  contractions  has  a  length  of  16  feet.  The  head 
as  measured  on  the  weir  is  2  feet  and  the  depth  of  the  channel  of  approach  below 
the  sill  of  the  weir  is  10  feet.  Find  the  discharge. 


Therefore  C  =  3-28. 

Approximately,  Q  =  3-28  2^.16 

=  148  cubic  feet  per  second. 

1  4.ft»^i 
The  velocity  v  =  -rz  —  T«  =  *77  '*•  Per  sec-> 

-Llj  X  JLu 

and  1^2=.  0147  feet. 

A  second  approximation  to  Q  is,  therefore, 

Q  =  3-28  (2-0147)*.  16 

=  150  cubic  feet  per  second. 

A  third  value  for  Q  can  be  obtained,  but  the  approximation  is  sufficiently  near 
for  all  practical  purposes. 

In  this  case  the  error  in  neglecting  the  velocity  of  approach  altogether,   is 
probably  less  than  the  error  involved  in  taking  m  as  0*4099. 


FLOW   OVER   WEIRS  109 

85.  Time  required  to  lower  the  water  in  a  reservoir  a 
given  distance  by  means  of  a  weir. 

A  reservoir  has  a  weir  of  length  L  feet  made  in  one  of  its  sides, 
and  having  its  sill  H  feet  below  the  original  level  of  the  water  in 
the  reservoir. 

It  is  required  to  find  the  time  necessary  for  the  water  to  fall  to 
a  level  H0  feet  above  the  sill  of  the  weir.  It  is  assumed  that  the 
area  of  the  reservoir  is  so  large  that  the  velocity  of  the  water  as 
it  approaches  the  weir  may  be  neglected. 

When  the  surface  of  the  water  is  at  any  height  h  above  the  sill 
the  flow  in  a  time  dt  is 


Let  A  be  the  area  of  the  water  surface  at  this  level  and  dh  the 
distance  the  surface  falls  in  time  dt. 

Then,  CU$dt  =  A.dh, 

Adh 
and  ot  =  --  «  . 

CL&* 

The  time   required  for  the  surface  to  fall  (H-H0)   feet  is, 
therefore, 

I     H  Adh 


The  coefficient  C  may  be  supposed  constant  and  equal  to  3'34. 
If  then  A  is  constant 

.          dh 

~ 


=  2A/J_     J_ 
CLWHo      N/H 


To  lower  the  level  to  the  sill  of  the  weir,  H0  must  be  made 
equal  to  0  and  t  is  then  infinite. 

That  is,  on  the  assumptions  made,  the  surface  of  the  water 
never  could  be  reduced  to  the  level  of  the  sill  of  the  weir.  The 
time  taken  is  not  actually  infinite  as  the  water  in  the  reservoir  is 
not  really  at  rest,  but  has  a  small  velocity  in  the  direction  of  the 
weir,  which  causes  the  time  of  emptying  to  be  less  than  that 
given  by  the  above  formula.  But  although  the  actual  time  is 
not  infinite,  it  is  nevertheless  very  great. 

9  A 

When  Ho  is  iH,  t=7^FF- 

OLvH 

When  Ho  is  TVH,  t  =  ^jj=. 

So  that  it  takes  three  times  as  long  for  the  water  to  fall  from 
iH  to  TVH  as  from  H  to  iH. 


110  HYDRAULICS 

Example  1.  A  reservoir  has  an  area  of  60,000  sq.  yards.  A  weir  10  feet  long 
has  its  sill  2  feet  below  the  surface.  Find  the  time  required  to  reduce  the  level  of 
the  water  1'  11". 

H0  =  Ty,         H  =  2'. 

Therefore  t=*'™  (3-46-0-708), 


=  89,000  sees. 
=  24-7  hours. 

So  that,  neglecting  velocity  of  approach,  there  will  be  only  one  inch  of  water  on 
the  weir  after  24  hours. 

Example  2.  To  find  in  the  last  example  the  discharge  from  the  reservoir  in 
15  hours. 

O  A        /I  1     \ 

Therefore  54,000  =  £-=-  (  —= -=  \  . 

From  which  ^/H^  0-421, 

H0= 0-176  feet. 
The  discharge  is,  therefore, 

(2  -  0-176)  540,000  cubic  feet 
=  984,960  cubic  feet. 

EXAMPLES. 

(1)  A  weir  is  100  feet  long  and  the  head  is  9  inches.   Find  the  discharge 
in  c.  ft.  per  minute.     C  =  3'34. 

(2)  The  discharge  through   a   sharp-edged  rectangular  weir  is   500 
gallons  per  minute,  and  the  still  water  head  is  2|  inches.   Find  the  effective 
length  of  the  weir,     m  =  '43. 

(3)  A  weir  is  15  feet  long  and  the  head  over  the  crest  is  15  inches. 
Find  the  discharge.     If  the  velocity  of  approach  to  this  weir  were  5  feet 
per  second,  what  would  be  the  discharge  ? 

(4)  Deduce  an  expression  for  the  discharge  through  a  right-angled 
triangular  notch.     If  the  head  over  apex  of  notch  is  12  ins.,  find  the 
discharge  in  c.  ft.  per  sec. 

(5)  A   rectangular  weir  is  to  discharge   10,000,000  gallons  per   day 
(1  gallon  =  10  Ibs.),  with  a  normal  head  of  15  ins.     Find  the  length  of  the 
weir.     Choose  a  coefficient,  stating  for  what  kind  of  weir  it  is  applicable, 
or  take  the  coefficient  C  as  3'33. 

(6)  What  is  the  advantage  in  gauging,  of  using  a  weir  without  end 
contractions  ? 

(7)  Deduce  Francis'  formula  by  means  of  the  Thomson  principle  of 
similarity. 

Apply  the  formula  to  calculate  the  discharge  over  a  weir  10  feet  wide 
under  a  head  of  1-2  feet,  assuming  one  end  contraction,  and  neglecting  the 
effect  of  the  velocity  of  approach. 


FLOW   OVER   WEIRS  111 

(8)  A  rainfall  of  ^  inch  per  hour  is  discharged  from  a  catchment  area 
of  5  square  miles.     Find  the  still  water  head  when  this  volume  flows  over 
a  weir  with  free  overfall  30  feet  in  length,  constructed  in  six  bays,  each 
5  feet  wide,  taking  0'415  as  Bazin's  coefficient. 

(9)  A  district  of  6500  acres  (1  acre =43,560  sq.  ft.)  drains  into  a  large 
storage  reservoir.     The  maximum  rate  at  which  rain  falls  in  the  district  is 
2  ins.  in  24  hours.     When  rain  falls  after  the  reservoir  is  full,  the  water 
requires  to  be  discharged  over  a  weir  or  bye-wash  which  has  its  crest  at 
the  ordinary  top-water  level  of  the  reservoir.     Find  the  length  of  such  a 
weir  for  the  above  reservoir,  under  the  condition  that  the  water  in  the 
reservoir  shall  never  rise  more  than  18  ins.  above  its  top-water  level. 

The  top  of  the  weir  may  be  supposed  flat  and  about  18  inches  wide 
(see  Table  XI). 

(10)  Compare  rectangular  and  V  notches  in  regard  to  accuracy  and 
convenience  when  there  is  considerable  variation  in  the  flow. 

In  a  rectangular  notch  50"  wide  the  still  water  surface  level  is  15"  above 
the  sill. 

If  the  same  quantity  of  water  flowed  over  a  right-angled  V  notch,  what 
would  be  the  height  of  the  still  water  surface  above  the  apex  ? 

If  the  channels  are  narrow  how  would  you  correct  for  velocity  of 
approach  in  each  case?  Lon.  Un.  1906. 

(11)  The  heaviest  daily  record  of  rainfall  for  a  catchment  area  was 
found  to  be  42*0  million  gallons.     Assuming  two-thirds  of  the  rain  to  reach 
the  storage  reservoir  and  to  pass  over  the  waste  weir,  find  the  length  of 
the  sill  of  the  waste  weir,  so  that  the  water  shall  never  rise  more  than  two 
feet  above  the  sill. 

(12)  A  weir  is  300  yards  long.     What  is  the  discharge  when  the  head 
is  4  feet  ?     Take  Bazin's  coefficient 

AnK     '00984 
m  =  '405  +  — -= —  . 
h 

(13)  Suppose  the  water  approaches  the  weir  in  the  last  question  in  a 
channel  8'  6"  deep  and  500  yards  wide.     Find  by  approximation  the  dis- 
charge, taking  into  account  the  velocity  of  approach. 

(14)  The  area  of  the  water  surface  of  a  reservoir  is  20,000  square 
yards.     Find  the  time  required  for  the  surface  to  fall  one  foot,  when  the 
water  discharges  over  a  sharp-edged  weir  5  feet  long  and  the  original  head 
over  the  weir  is  2  feet. 

(15)  Find,  from  the  following  data,  the  horse-power  available  in  a  given 
waterfall  :— 

Available  height  of  fall  120  feet. 

A  rectangular  notch  above  the  fall,  10  feet  long,  is  used  to  measure 
the  quantity  of  water,  and  the  mean  head  over  the  notch  is  found  to  be 
15  inches,  when  the  velocity  of  approach  at  the  point  where  the  head 
is  measured  is  100  feet  per  minute.  Lon.  Un.  1905. 


CHAPTER  V. 


FLOW  THROUGH  PIPES. 

86.     Resistances  to  the  motion  of  a  fluid  in  a  pipe. 

When  a  fluid  is  made  to  flow  through  a  pipe,  certain  resistances 
are  set  up  which  oppose  the  motion,  and  energy  is  consequently 
dissipated.  Energy  is  lost,  by  friction,  due  to  the  relative  motion 
of  the  water  and  the  pipe,  by  sudden  enlargements  or  contractions 
of  the  pipe,  by  sudden  changes  of  direction,  as  at  bends,  and  by 
obstacles,  such  as  valves  which  interfere  with  the  free  flow  of  the 
fluid. 

It  will  be  necessary  to  consider  these  causes  of  the  loss  of 
energy  in  detail. 

Loss  of  head.  Before  proceeding  to  do  so,  however,  the  student 
should  be  reminded  that  instead  of  loss  of  energy  it  is  convenient 
to  speak  of  the  loss  of  head. 

It  has  been  shown  on  page  39  that  the  work  that  can  be 
obtained  from  a  pound  of  water,  at  a  height  z  above  datum, 
moving  with  a  velocity  v  feet  per  second,  and  at  a  pressure  head 

79  /D         V^ 

— ,  is   —  +  —  +  z  foot  pounds. 
w*       w     2g 

If  now  water  flows  along  a  pipe  and,  due  to  any  cause,  h  foot 
pounds  of  work  are  lost  per  pound,  the  available  head  is  clearly 
diminished  by  an  amount  h. 

In  Fig.  86  water  is  supposed  to  be  flowing  from  a  tank  through 
a  pipe  of  uniform  diameter  and  of  considerable  length,  the  end  B 
being  open  to  the  atmosphere. 


D 


Fig.  86.     Loss  of  head  by  friction  in  a  pipe. 


FLOW  THROUGH  PIPES  113 

Let  —  be  the  head  due  to  the  atmospheric  pressure. 

Then  if  there  were  no  resistances  and  assuming  stream  line 
flow,  Bernoulli's  equation  for  the  point  B  is 

,  Pa  ,  v*      „       pa 

ZB  +  —  +  77-  =  Zip  +  —  , 
w      2g  w  ' 


from  which  ^  -  ZP  -  ZB  -  H, 

or  VE  =  \/2#H. 

The  whole  head  H  above  the  point  B  has  therefore  been 
utilised  to  give  the  kinetic  energy  to  the  water  leaving  the  pipe  at 
B.  Experiment  would  show,  however,  that  the  mean  velocity  of 
the  water  would  have  some  value  v  less  than  VB,  and  the  kinetic 

energy  would  be  ~-  . 


B  „ 

A  head  h  =  -=  --  -r-  =  H  - 


YB2 
-= 

Zg      2g 

has  therefore  been  lost  in  the  pipe. 

By  carefully  measuring  H,  the  diameter  of  the  pipe  d,  and  the 
discharge  Q  in  a  given  time,  the  loss  of  head  h  can  be  determined. 

For  v  =  —-£-—  , 

(H 

Q2 

and  therefore  h  =  H  --  —  — 


The  head  h  clearly  includes  all  causes  of  loss  of  head,  which, 
in  this  case,  are  loss  at  the  entrance  of  the  pipe  and  loss  by 
friction. 

87.     Loss  of  head  by  friction. 

Suppose  tubes  1,  2,  3  are  fitted  into  the  pipe  AB,  Fig.  86,  at 
equal  distance  apart,  and  with  their  lower  ends  flush  with  the  inside 
of  the  pipe,  and  the  direction  of  the  tube  perpendicular  to  the 
direction  of  flow.  If  flow  is  prevented  by  closing  the  end  B  of  the- 
pipe,  the  water  would  rise  in  all  the  tubes  to  the  level  of  the  water- 
in  the  reservoir. 

Further,  if  the  flow  is  regulated  at  B  by  a  valve  so  that  thfr 
mean  velocity  through  the  pipe  is  v  feet  per  second,  a  permanent 
regime  being  established,  and  the  pipe  is  entirely  full,  the  mean 
velocity  at  all  points  along  the  pipe  will  be  the  same  ;  and  there- 
fore, if  between  the  tank  and  the  point  B  there  were  no  resistances 
offered  to  the  motion,  and  it  be  assumed  that  all  the  particles 
L.  H.  8 


114  HYDRAULICS 

have  a  velocity  equal  to  the  mean  velocity,  the  water  would  again 

rise  in  all  the  tubes  to  the  same  height,  but  now  lower  than  the 

0i 
surface  of  the  water  in  the  tank  by  an  amount  equal  to  ^-  . 

It  is  found  by  experiment,  however,  that  the  water  does  not 
rise  to  the  same  height  in  the  three  tubes,  but  is  lower  in  2  than 
in  1  and  in  3  than  in  2  as  shown  in  the  figure.  As  the  fluid  moves 
along  the  pipe  there  is,  therefore,  a  loss  of  head. 

The  difference  of  level  h2  of  the  water  in  the  tubes  1  and  2  is 
called  the  head  lost  by  friction  in  the  length  of  pipe  1  2.  In  any 
length  I  of  the  pipe  the  loss  of  head  is  h. 

This  head  is  not  wholly  lost  simply  by  the  relative  movement 
of  the  water  and  the  surface  of  the  pipe,  as  if  the  water  were 
a  solid  body  sliding  along  the  pipe,  but  is  really  the  sum  of  the 
losses  of  energy,  by  friction  along  the  surface,  and  due  to  relative 
motions  in  the  mass  of  water. 

It  will  be  shown  later  that,  as  the  water  flows  along  the  pipe, 
there  is  relative  motion  between  consecutive  filaments  in  the  pipe, 
and  that,  when  the  velocity  is  above  a  certain  amount,  the  water 
has  a  sinuous  motion  along  the  pipe.  Some  portion  of  this  head  h2 
is  therefore  lost,  by  the  relative  motion  of  the  filaments  of  water, 
and  by  the  eddy  motions  which  take  place  in  the  mass  of  the 
water. 

When  the  pipe  is  uniform  the  loss  of  head  is  proportional 
to  the  length  of  the  pipe,  and  the  line  CB,  drawn  through  the  tops 
of  the  columns  of  water  in  the  tubes  and  called  the  hydraulic 
gradient,  is  a  straight  line. 

It  should  be  noted  that  along  CB  the  pressure  is  equal  to  that 
of  the  atmosphere. 

88.     Head  lost  at  the  entrance  to  the  pipe. 

For  a  point  E  just  inside  the  pipe,  Bernoulli's  equation  is 

2 

—  +  Q-  +  head  lost  at  entrance  to  the  pipe  =  h&  +  —  , 

—  being  the  absolute  pressure  head  at  E. 

The  head  lost  at  entrance  has  been  shown  on  page  70  to  be 


,  , 

about  -^  —  ,  and  therefore, 


pE     pa     -, 
—  — 


. 
w      w  2(7 

That  is,  the  point  C  on  the  hydraulic  gradient  vertically  above 

1*50* 

E,  is  -—  below  the  surface  FD. 


FLOW   THROUGH   PIPES 


115 


If  the  pipe  is  bell-mouthed,  there  will  be  no  head  lost  at  entrance, 
and  the  point  C  is  a  distance  equal  to  —•  below  the  surface. 

89.     Hydraulic  gradient  and  virtual  slope. 

The  line  CB  joining  the  tops  of  the  columns  of  water  in  the 
tube,  is  called  the  hydraulic  gradient,  and  the  angle  i  which  it 
makes  with  the  horizontal  is  called  the  slope  of  the  hydraulic 
gradient,  or  the  virtual  slope.  The  angle  i  is  generally  small,  and 

sin  i  may  be  taken  therefore  equal  to  i,  so  that  j  =  i. 
In  what  follows  the  virtual  slope  -y  is  denoted  by  i. 

i 

More  generally  the  hydraulic  gradient  may  be  defined  as  the 
line,  the  vertical  distance  between  which  and  the  centre  of  the 
pipe  gives  the  pressure  head  at  that  point  in  the  pipe.  This  line 
will  only  be  a  straight  line  between  any  two  points  of  the  pipe, 
when  the  head  is  lost  uniformly  along  the  pipe. 

If  the  pressure  head  is  measured  above  the  atmospheric 
pressure,  the  hydraulic  gradient  in  Fig.  87  is  AD,  but  if  above 
zero,  AiDi  is  the  hydraulic  gradient,  the  vertical  distance  between 

r>  144 
AD  and  AiDi  being  equal  to  -     — ,  pa  being  the   atmospheric 

pressure  per  sq.  inch. 


Fig.  87.     Pipe  rising  above  the  Hydraulic  Gradient. 

If  the  pipe  rises  above  the  hydraulic  gradient  AD,  as  in  Fig.  87, 
the  pressure  in  the  pipe  at  C  will  be  less  than  that  of  the  atmosphere 
by  a  head  equal  to  CE.  If  the  pipe  is  perfectly  air-tight  it  will 
act  as  a  siphon  and  the  discharge  for  a  given  length  of  pipe  will 
not  be  altered.  But  if  a  tube  open  to  the  atmosphere  be  fitted  at 

8—2 


116 


HYDRAULICS 


the  highest  point,  the  pressure  at  C  is  equal  to  the  atmospheric 
pressure,  and  the  hydraulic  gradient  will  be  now  AC,  and  the  flow 
will  be  diminished,  as  the  available  head  to  overcome  the  resist- 
ances between  B  and  C,  and  to  give  velocity  to  the  water,  will  only 
be  CF,  and  the  part  of  the  pipe  CD  will  not  be  kept  full. 

In  practice,  although  the  pipe  is  closed  to  the  atmosphere,  yet 
air  will  tend  to  accumulate  and  spoil  the  siphon  action. 

As  long  as  the  .point  C  is  below  the  level  of  the  water  in  the 
reservoir,  water  will  flow  along  the  pipe,  but  any  accumulation  of 
air  at  C  tends  to  diminish  the  flow.  In  an  ordinary  pipe  line  it  is 
desirable,  therefore,  that  no  point  in  the  pipe  should  be  allowed  to 
rise  above  the  hydraulic  gradient. 

90.  Determination  of  the  loss  of  head  due  to  friction. 
Reynolds'  apparatus. 

Fig.  88  shows  the  apparatus  as  used  by  Professor  Reynolds*  for 
determining  the  loss  of  head  by  friction  in  a  pipe.  , 


Fig.  88.     Reynolds'  apparatus  for  determining  loss  of  head  by  friction  in  a  pipe. 

A  horizontal  pipe  AB,  16  feet  long,  was  connected  to  the  water 
main,  a  suitable  regulating  device  being  inserted  between  the 
main  and  the  pipe. 

At  two  points  5  feet  apart  near  the  end  B,  and  thus  at  a  distance 
sufficiently  removed  from  the  point  at  which  the  water  entered 
the  pipe,  that  any  initial  eddy  motions  might  be  destroyed  and  a 
steady  regime  established,  two  holes  of  about  1  mm.  diameter  were 
pierced  into  the  pipe  for  the  purpose  of  gauging  the  pressure,  at 
these  points  of  the  pipe. 

Short  tubes  were  soldered  to  the  pipe,  so  that  the  holes 
communicated  with  these  tubes,  and  these  were  connected  by 

*  Phil.  Trans.  1883,  or  Vol.  n.  Scientific  Papers,  Reynolds. 


FLOW   THROUGH   PIPES  117 

indiarubber  pipes  to  the  limbs  of  a  siphon  gauge  G-,  made  of  glass, 
and  which  contained  mercury  or  bisulphide  of  carbon.  Scales 
were  fixed  behind  the  tubes  so  that  the  height  of  the  columns 
in  each  limb  of  the  gauge  could  be  read. 

For  very  small  differences  of  level  a  cathetometer  was  used  *. 
When  water  was  made  to  flow  through  the  pipe,  the  difference  in 
the  heights  of  the  columns  in  the  two  limbs  of  the  siphon  measured 
the  difference  of  pressure  at  the  two  points  A  and  B  of  the  pipe, 
and  thus  measured  the  loss  of  head  due  to  friction. 

If  s  is  the  specific  gravity  of  the  liquid,  and  H  the  difference 
in  height  of  the  columns,  the  loss  of  head  due  to  friction  in  feet  of 
water  is  h  =  H  (s  -  1). 

The  quantity  of  water  flowing  in  a  time  t  was  obtained  by 
actual  measurement  in  a  graduated  flask. 

Calling  v  the  mean  velocity  in  the  pipe  in  feet  per  second,  Q 
the  discharge  in  cubic  feet  per  second,  and  d  the  diameter  of  the 
pipe  in  feet, 


The  loss  of  head  at  different  velocities  was  carefully  measured, 
and  the  law  connecting  head  lost  in  a  given  length  of  pipe,  with 
the  velocity,  determined. 

The  results  obtained  by  Keynolds,  and  others,  using  this 
method  of  experimenting,  will  be  referred  to  later. 

91.  Equation  of  flow  in  a  pipe  of  uniform  diameter 
and  determination  of  the  head  lost  due  to  friction. 

.Let  8Z  be  the  length  of  a  small  element  of  pipe  of  uniform 
diameter,  Fig.  89. 

A 


C 

Fig.  89. 

Let  the  area  of  the  transverse  section  be  w,  P  the  length  of 
the  line  of  contact  of  the  water  and  the  surface  on  this  section,  or 
the  wetted  perimeter,  a  the  inclination  of  the  pipe,  p  the  pressure 
per  unit  area  on  AB,  and  p  —  dp  the  pressure  on  CD. 
*  p.  258,  Vol.  i.  Scientific  Papers,  Keynolds. 


118  HYDRAULICS 

Let  v  be  the  mean  velocity  of  the  fluid,  Q  the  flow  in  cubic 
feet  per  second,  and  w  the  weight  of  one  cubic  foot  of  the  fluid. 
The  work  done  by  gravity  as  the  fluid  flows  from  AB  to  CD 

=  Qw  .  dz  —  <*>  .  v  .  w  .  dz. 

The  work  done  on  ABCD  by  the  pressure  acting  upon  the  area 
AB 

=  p .  w  .  v  ft.  Ibs.  per  sec. 

The  work  done  by  the  pressure  acting  upon  CD  against  the 
flow 

=  (p  —  dp)  .  o> .  v  ft.  Ibs.  per  sec. 

The  frictional  force  opposing  the  motion  is  proportional  to  the 
area  of  the  wetted  surface  and  is  equal  to  F .  P .  ol,  where  F  is  some 
coefficient  which  must  be  determined  by  experiment  and  is  the 
frictional  force  per  unit  area.  The  work  done  by  friction  per  sec. 
is,  therefore,  F  .  P .  bl .  v. 

The  velocity  being  constant,  the  velocity  head  is  the  same  at 
both  sections,  and  therefore,  applying  the  principle  of  the  con- 
servation of  energy, 

p.<o.v  +  <a.v  .w  .dz  =  (p-  dp)  to  .  v  +  F .  P .  9Z .  v. 

Therefore  w  .  w  .  dz  =  -  dp .  w  +  F  .  P .  3Z, 

,         dp     F.P.dl 

or  dz  =  — -  +  -          -  . 

w         w .  to 

Integrating  this  equation  between  the  limits  of  z  and  Zi,  p  and 
Pi  being  the  corresponding  pressures,  and  Z  the  length  of  the  pipe, 

p1     p     F  .  P  I 

z  —  Zi  =  —  ——  +  -        — . 

W        W  W       to 

Therefore,  £  +  *  =  &  +  z1+  ^  1 . 

W  W  W      to 

The  quantity  -  -  is  equal  to  h  of  equation  (1),  page  52,  and  is 

the  loss  of  head  due  to  friction.  The  head  lost  by  friction  is 
therefore  proportional  to  the  area  of  the  wetted  surface  of  the  pipe 
PZ,  and  inversely  proportional  to  the  cross  sectional  area  of  the 
pipe  and  to  the  density  of  the  fluid. 

92.     Hydraulic  mean  depth. 

The  quantity  p  is  called  the  hydraulic  radius,  or  the  hydraulic 

mean  depth. 

If  then  this  quantity  is  denoted  by  m,  the  head  h  lost  by 
friction,  is 

FZ 


h  = 


w  .m 


FLOW  THROUGH   PIPES  119 

The  quantity  F,  which  has  been  called  above  the  friction  per 
unit  area,  is  found  by  experiments  to  vary  with  the  density, 
viscosity,  and  velocity  of  the  fluid,  and  with  the  diameter  and 
roughness  of  the  internal  surface  of  the  pipe. 

In  Hydraulics,  the  fluid  considered  is  water,  and  any  variations 
in  density  or  viscosity,  due  to  changes  of  temperature,  are  generally 
negligible.  F,  therefore,  may  be  taken  as  proportional  to  the 
density,  or  to  the  weight  w  per  cubic  foot,  to  the  roughness  of  the 
pipe,  and  as  some  function,  f(v)  of  the  mean  velocity,  and  f(d)  of 
the  diameter  of  the  pipe. 

Then,  h 


m 

in  which  expression  p.  may  be  called  the  coefficient  of  friction. 

It  will  be  seen  later,  that  the  mean  velocity  v  is  different  from 
the  relative  velocity  u  of  the  water  and  the  surface  of  the  pipe, 
and  it  probably  would  be  better  to  express  F  as  a  function  of  u, 
but  as  u  itself  probably  varies  with  the  roughness  of  the  pipe  and 
with  other  circumstances,  and  cannot  directly  be  determined,  it 
simplifies  matters  to  express  F,  and  thus  h,  as  a  function  of  v. 

93.    Empirical  formulae  for  loss  of  head  due  to  friction. 

The  difficulty  of  correctly  determining  the  exact  value  of 
fW  fWy  nas  led  to  the  use  of  empirical  formulae,  which  have 
proved  of  great  practical  service,  to  express  the  head  h  in  terms  of 
the  velocity  and  the  dimensions  of  the  pipe. 

The  simplest  formula  assumes  that  the  friction  simply  varies  as 
the  square  of  the  velocity,  and  is  independent  of  the  diameter  of 
the  pipe,  or  f(v)  f(d)  =  av2. 

mi,  -,      av2l  /-.N 

Then,  h  =  —  -  ..  .............................  (1), 

or  writing  ™  for  a, 


from  which  is  deduced  the  -well-known  Chezy  formula, 


n     /* 

t>  =  Cy  m. j, 


or  v  =  C  \/mi. 

Another  form  in  which  formula  (1)  is  often  found  is 

h-f^L 

1 1  —  ^r  , 

2g  m' 


120 


HYDRAULICS 


or  since  m  =  -7  for  a  circular  pipe  full  of  water, 


h 


.(3), 


in  wliicli  for  a  of  (1)  is  substituted  £- . 

The  quantity  Zg  was  introduced  by  Weisbach  so  that  h  is 
expressed  in  terms  of  the  velocity  head. 

Adopting  either  of  these  forms,  the  values  of  the  coefficients  C 
and  /  are  determined  from  experiments  on  various  classes  of  pipes. 


It  should  be  noticed  that  C  = 


_     /^ 

-V  ?' 


Values  of  these  constants  are  shown  in  Tables  XII  to  XIY  for 
different  kinds  and  diameters  of  pipes  and  different  velocities. 


TABLE  XII. 

Values  of  C  in  the  formula  v  =  C  *Jmi  for  new  and  old  cast-iron 
pipes. 


New  cast-iron  pipes 

Old  cast-iron  pipes 

Velocities  in  ft.  per  second 

1 

3 

6 

10 

1 

3 

6 

10 

Diameter  of  pipe 

3" 

95 

98 

100 

102 

63 

68 

71 

73 

6" 

96 

101 

104 

106 

69 

74 

77 

79 

9" 

98 

105 

109 

112 

73 

78 

80 

84 

12" 

100 

108 

112 

117 

77 

82 

85 

88 

15" 

102 

110 

117 

122 

81 

86 

89 

91 

18" 

105 

112 

119 

125 

86 

91 

94 

97 

24" 

111 

120 

126 

131 

92 

98 

101 

104 

30" 

118 

126 

131 

136 

98 

103 

106 

109 

36" 

124 

131 

136 

140 

103 

108 

111 

114 

42" 

130 

136 

140 

144 

105 

111 

114 

117 

48" 

135 

141 

145 

148 

106 

112 

115 

118 

60" 

142 

147 

150 

152 

For  method  of  determining  the  values  of  C  given  in  the  tables, 
see  page  102. 

On  reference  to  these  tables,  it  will  be  seen,  that  C  and  /  are 
by  no  means  constant,  but  vary  very  considerably  for  different 
kinds  of  pipes,  and  for  different  values  of  the  velocity  in  any 
given  pipe. 


FLOW  THROUGH   PIPES 


121 


The  fact  that  C  varies  with  the  velocity,  and  the  diameter  of 
the  pipe,  suggests  that  the  coefficient  C  is  itself  some  function  of 
the  velocity  of  flow,  and  of  the  diameter  of  the  pipe,  and  that 
jjf(v)  f(d)  does  not,  therefore,  equal  CLV*. 

TABLE  XIII. 

Values  of  /  in  the  formula 


2gd 


New  cast-iron  pipes 

Old  cast-iron  pipes 

Velocities  in 
ft.  per  second 

1 

3 

6 

10 

1 

3 

6 

10 

Diam.  of  pipe 

3" 

•0071 

•0067 

•0064 

•0062 

•0152 

•0139 

•0128 

•0122 

6" 

•007 

•0063 

•006 

•0057 

•0135 

•0117 

•0108 

•0103 

9" 

•0067 

•0058 

•0055 

•0051 

•0122 

•0105 

•010 

•0092 

12" 

•0064 

•0056 

•0051 

•0048 

•0108 

•0096 

•0089 

•0084 

15" 

•0062 

•0053 

•0048 

•0043 

•0099 

•0087 

•0081 

•0078 

18" 

•0058 

•0051 

•0045 

•0041 

•0087 

•0078 

•0073 

•0069 

24" 

•0053 

•0045 

•0040 

•0037 

•0076 

•0067 

•0063 

•0060 

30" 

•0046 

•0040 

•0037 

•0035 

•0067 

•0061 

•0057 

•0055 

36" 

•0042 

•0037 

•0035 

•0033 

•0061 

•0056 

•0052 

•0050 

42" 

•0038 

•0035 

•0033 

•0031 

•0058 

•0052 

•005 

•0048 

48" 

•0036 

•0032 

•0031 

•0029 

•0057 

•0051 

•0049 

•0046 

60" 

•0032 

•0030 

•0029 

•0028 

TABLE  XIV. 

Values  of  C  in  the  formula  v  =  C  *Jmi  for  steel  riveted  pipes. 


Velocities  in  ft.  per  second 

1 

3 

5 

10 

Diameter  of  pipe 

3" 

81 

86 

89 

92 

11" 

92 

102 

107 

115 

llf 
15" 

93 

109 

99 
112 

102 
114 

105 
117 

38" 

113 

113 

113 

113 

42" 

102 

106 

108 

111 

48" 

105 

105 

105 

105 

72"* 

110 

110 

111 

111 

72" 

93 

101 

105 

110 

103" 

114 

109 

106 

104 

*  See  pages  124  and  137. 


122  HYDRAULICS 

94.     Formula  of  Darcy. 

In  1857  Darcy*  published  an  account  of  a  series  of  experiments 
on  flow  of  water  in  pipes,  previous  to  the  publication  of  which,  it 
had  been  assumed  by  most  writers  that  the  friction  and  consequently 
the  constant  C  was  independent  of  the  nature  of  the  wetted  surface 
of  the  pipe  (see  page  232).  He,  however,  showed  by  experiments 
upon  pipes  of  various  diameters  and  of  different  materials, 
including  wrought  iron,  sheet  iron  covered  with  bitumen,  lead, 
glass,  and  new  and  old  cast-iron,  that  the  condition  of  the  internal 
surface  was  of  considerable  importance  and  that  the  resistance  was 
by  no  means  independent  of  it. 

He  also  investigated  the  influence  of  the  diameter  of  the  pipe 
upon  the  resistance.  The  results  of  his  experiments  he  expressed 
by  assuming  the  coefficient  a  in  the  formula 

7          Q>1          2 

h  =  — .  tr 

m 

was  of  the  form  a  =  a  +  - 

T 

r  being  the  radius  of  the  pipe. 

For  new  cast-iron,  and  wrought-iron  pipes  of  the  same 
roughness,  Darcy's  values  of  a  and  ft  when  transferred  to  English 
units  are, 

a  =  0-000077, 
ft  =  0-000003235. 

For  old  cast-iron  pipes  Darcy  proposed  to  double  these  values. 
Substituting  the  diameter  d  for  the  radius  r,  and  doubling  /?,  for 
new  pipes, 

=    0-000077 


12d+l 

Substituting  for  m  its  value  -^ ,  and  multiplying  and  dividing 
by  20, 


For  old  cast-iron  pipes, 

h  =  0-00001294 


d      /  m 

.J  ........................  (7). 

Eecherches  Experlmentales. 


FLOW  THROUGH   PIPES  123 


As  the  student  cannot  possibly  retain,  without  unnecessary 
labour,  values  of  /  and  C  for  different  diameters  it  is  convenient 
to  remember  the  simple  forms, 

f=-°°5(1  + 

for  new  pipes,  and 


for  old  pipes. 

According  to  Darcy,  therefore,  the  coefficient  C  in  the  Chezy 
formula  varies  only  with  the  diameter  and  roughness  of  the  pipe. 

The  values  of  C  as  calculated  from  his  experimental  results,  for 
some  of  the  pipes,  were  practically  constant  for  all  velocities,  and 
notably  for  those  pipes  which  had  a  comparatively  rough  internal 
surface,  but  for  smooth  pipes,  the  value  of  C  varied  from  10  to 
20  per  cent,  for  the  same  pipe  as  the  velocity  changed.  The 
experiments  of  other  workers  show  the  same  results. 

The  assumption  that  pf(v)  f(d)  -  at;2  in  which  a  is  made  to 
vary  only  with  the  diameter  and  roughness,  or  in  other  words,  the 
assumption  that  h  is  proportional  to  v2  is  therefore  not  in  general 
justified  by  experiments. 

95.  As  stated  above,  the  formulae  given  must  be  taken  as 
purely  empirical,  and  though  by  the  introduction  of  suitable 
constants  they  can  be  made  to  agree  with  any  particular  experi- 
ment, or  even  set  of  experiments,  yet  none  of  them  probably 
expresses  truly  the  laws  of  fluid  friction. 

The  formula  of  Chezy  by  its  simplicity  has  found  favour,  and 
it  is  likely,  that  for  some  time  to  come,  it  will  continue  to  be  used, 
either  in  the  form  v  =  C  vW,  or  in  its  modified  form 


~  -  ^ 

2gd 

In  making  calculations,  values  of  C  or  /,  which  most  nearly  suit 
any  given  case,  can  be  taken  from  the  tables. 

96.     Variation  of  C  in  the  formula  v  =  C  */m~i  with  service. 

It  should  be  clearly  borne  in  mind,  however,  that  the  dis- 
charging capacity  of  a  pipe  may  be  considerably  diminished  after 
a  few  years'  service. 

Darcy's  results  show  that  the  loss  of  head  in  an  old  pipe  may 
be  double  that  in  a  new  one,  or  since  the  velocity  v  is  taken  as 


124  HYDRAULICS 

proportional  to  the  square  root  of  Ji,  the  discharge  of  the  old  pipe 
for  the  same  head  will  be  —^  times  that  of  the  new  pipe,  or  about 

30  per  cent.  less. 

An  experiment  by  Sherman*  on  a  36-inch  cast-iron  main  showed 
that  after  one  year's  service  the  discharge  was  diminished  by 
23  per  cent.,  but  a  second  year's  service  did  not  make  any  further 
alteration. 

Experiments  by  Kuichlingt  on  a  36-inch  cast-iron  main  showed 
that  the  discharge  during  four  years  diminished  36  per  cent.,  while 
experiments  by  Fitzgerald  J  on  a  cast-iron  main,  coated  with  tar, 
which  had  been  in  use  for  16  years,  showed  that  cleaning  increased 
the  discharge  by  nearly  40  per  cent.  Fitzgerald  also  found  that 
the  discharge  of  the  Sudbury  aqueduct  diminished  10  per  cent,  in 
one  year  due  to  accumulation  of  slime. 

The  experiments  of  Marx,  Wing,  and  Hoskins  §  on  a  72-inch  steel 
main,  when  new,  and  after  two  years'  service,  showed  that  there 
had  been  a  change  in  the  condition  of  the  internal  surface  of  the 
pipe,  and  that  the  discharge  had  diminished  by  10  per  cent,  at  low 
velocities  and  about  5  per  cent,  at  the  higher  velocities. 

If,  therefore,  in  calculations  for  pipes,  values  of  C  or  /  are  used 
for  new  pipes,  it  will  in  most  cases  be  advisable  to  make  the  pipe 
of  such  a  size  that  it  will  discharge  under  the  given  head  at  least 
from  10  to  30  per  cent,  more  than  the  calculated  value. 

97.     Ganguillet  and  Kutter's  formula. 

Granguillet  and  Kutter  endeavoured  to  determine  a  form  for 
the  coefficient  C  in  the  Chezy  formula  v  =  C  \/W',  applicable 
to  all  forms  of  channels,  and  in  which  C  is  made  a  function  of  the 
virtual  slope  i,  and  also  of  the  diameter  of  the  pipe. 

They  gave  C  the  value, 

11-6  i  1'811  i  °'00281 

to  n 

=  ...............  ° 


This  formula  is  very  cumbersome  to  use,  and  the  value  of  the 
coefficient  of  roughness  n  for  different  cases  is  uncertain.  Tables 
have  however  been  prepared  which  considerably  facilitate  the  use 
of  the  formula. 

*  Trans.  Am.S.C.E.  Vol.  XLIV.  p.  85. 
t  Trans.  Am.S.  C.E.  Vol.  XLIV.  p.  56. 
$  Trans.  Am.S.C.E.  Vol.  XLIV.  p.  87. 
§  See  Table  No.  XIV. 


FLOW  THROUGH   PIPES  125 

Values  of  n  in  Ganguillet  and  Kutter's  formula. 
Wood  pipes  =  '01,  may  be  as  high  as  '015. 

Cast-iron  and  steel  pipes  =  'Oil,  „          „          *02. 

Grlazed  earthenware         =  '013. 

98.     Reynolds'  experiments  and  the  logarithmic  formula. 

The  formulae  for  loss  of  head  due  to  friction  previously  given 
have  all  been  founded  upon  a  probable  law  of  variation  of  h 
with  v,  but  no  rational  basis  for  the  assumptions  has  been  adduced. 

It  has  been  stated  in  section  93,  that  on  the  assumption  that  h 
varies  with  v2,  the  coefficient  C  in  the  formula 

'  m^ 

is  itself  a  function  of  the  velocity. 

The  experiments  and  deductions  of  Reynolds,  and  of  later 
workers,  throw  considerable  light  upon  this  subject,  and  show  that 
h  is  proportional  to  vn,  where  n  is  an  index  which  for  very  small 
velocities* — as  previously  shown  by  Poiseuille  by  experiments  on 
capillary  tubes — is  equal  to  unity,  and  for  higher  velocities  may 
have  a  variable  value,  which  in  many  cases  approximates  to  2. 

As  Darcy's  experiments  marked  a  decided  advance,  in  showing 
experimentally  that  the  roughness  of  the  wetted  surface  has  an 
effect  upon  the  loss  due  to  friction,  so  Reynolds'  work  marked 
a  further  step  in  showing  that  the  index  n  depends  upon  the  state 
of  the  internal  surface,  being  generally  greater  the  rougher  the 
surface. 

The  student  will  be  better  able  to  follow  Reynolds,  by  a  brief 
consideration  of  one  of  his  experiments. 

In  Table  XV  are  shown  the  results  of  an  experiment  made 
by  Reynolds  with  apparatus  as  illustrated  in  Fig.  88. 

In  columns   1   and  5  are  shown  the  experimental  values  of 

i  =  j,  and  v  respectively. 

The  curves,  Fig.  90,  were  obtained  by  plotting  v  as  abscissae 
and  i  as  ordinates. 

For  velocities  up  to  1*347  feet  per  second,  the  points  lie  very  close 
to  a  straight  line  and  i  is  simply  proportional  to  the  velocity,  or 

i  =  kiv  (11), 

Jci  being  a  coefficient  for  this  particular  pipe. 

Above  2  feet  per  second,  the  points  lie  very  near  to  a  continuous 
curve,  the  equation  to  which  is 

<  =  *»•  (12). 

*  Phil.  Tram.  1883. 


126 


HYDRAULICS 


Taking  logarithms, 


log  i  =  log  k  +  n  log  v. 


Curve  N?2  is  the  part  AB  of 


Carve  N?l  drawn  to  larger 
Scale.  y 


VeLodty. 

Fig.  90. 


The  curve,  Fig.  90  a,  was  determined  by  plotting  log  i  as 
ordinate  and  logv  as  abscissae.  Eeynolds  calls  the  lines  of  this 
figure  the  logarithmic  homologues. 

Calling  log  ij  y,  and  log  v,  x,  the  equation  has  the  form 


nx. 


which  is  an  equation  to  a  straight  line,  the  inclination  of  which  to 
the  axis  of  x  is 

0  =  tan"1  n. 


or 


n  =  tan  0. 


Further,  when  x  =  0,  y  =  &,  so  that  the  value  of  ~k  can  readily  be 
found  as  the  ordinate  of  the  line  when  x  or  \ogv  =  Q,  that  is, 
when  v  =  1. 

Up  to  a  velocity  of  1'37  feet  per  second,  the  points  lie  near  to 
a  line  inclined  at  45  degrees  to  the  axis  of  v,  and  therefore,  n  is 
unity,  or  as  stated  above,  i  =  kv. 

The  ordinate  when  'v  is  equal  to  unity  is  0'038,  so  that  for  the 
first  part  of  the  curve  k  =  '038,  and  i  =  '03&v. 


FLOW  THROUGH   PIPES 


127 


Above  the  velocity  of  2  feet  per  second  the  points  lie  about 
a  second  straight  line,  the  inclination  of  which  to  the  axis  of  v  is 

6  =  tan-1 170. 

Therefore  log  i  =  170  log  v  +  k. 

The  ordinate  when  v  equals  1  is  0*042,  so  that 

Je  =  0-042, 
and  i  =  0*042vr7°. 


-4-0 

-3-0 

-2-0 


-1-0 

-9 

--S 

_-7 
•6 

-5 


veLocUy 


-02 


•01 


\  Log  v 
•4    -5  •€  -7-8-91\0 


3      4     5    6  789W 


20 


Fig.  90  a.     Logarithmic  plottings  of  i  and  v  to  determine  the  index  n  in 
the  formula  for  pipes,  i=kvn. 

In  the  table  are  given  values  of  i  as  determined  experimentally 
and  as  calculated  from  the  equation  i  =  Jc .  vn. 

The  quantities  in  the  two  columns  agree  within  3  per  cent. 


123 


HYDRAULICS 


TABLE  XV. 

Experiment  on  Resistance  in  Pipes. 
Lead  Pipe.     Diameter  0'242".    Water  from  Manchester  Main. 


Slope  t=* 

k 

n 

Velocity  ft.  per 
second 

Experimental  value 

Calculated  from 

•0086 

•0092 

•038 

1 

•239 

•0172 

•0172 

•038 

1 

•451 

•0258 

•0261 

•038 

1 

•690 

•0345 

•0347 

•038 

1 

•914 

•0430 

•0421 

•038 

1 

1-109 

•0516 

•0512 

•038 

1 

1-349 

•0602 

... 

... 

1-482 

•0682 

... 

... 

1-573 

•0861 

1-671 

•1033 

... 

1-775 

•1206 

... 

1-857 

•1378 

•1352 

•042 

1-70 

1-987 

•1714 

•1610 

•042 

1-70 

2-203 

•3014 

•2944 

•042 

1-70 

3-141 

•4306 

•4207 

•042 

1-70 

3-93 

•8185 

•8017 

•042 

1-70 

5-66 

1-021 

1-033 

•042 

1-70 

6-57 

1-433 

1-476 

•042 

1-70 

8-11 

2-455 

2-404 

•042 

1-70 

10-79 

3-274 

3-206 

•042 

1-70 

12-79 

3-873 

3-899 

•042 

1-70 

14-29 

NOTE.     To  make  the  columns  shorter,  only  part  of  Reynolds'  results  are  given. 

99.     Critical  velocity. 

It  appears,  from  Reynolds'  experiment,  that  up  to  a  certain 
velocity,  which  is  called  the  Critical  Velocity,  the  loss  of  head  is 
proportional  to  v,  but  above  this  velocity  there  is  a  definite  change 
in  the  law  connecting  i  and  v. 

By  experiments  upon  pipes  of  different  diameters  and  the 
water  at  variable  temperatures,  Reynolds  found  that  the  critical 
velocity,  which  was  taken  as  the  point  of  intersection  of  the  two 
straight  lines,  was 

•0388P 


the  value  of  P  being 
P  = 


(13), 


1+  0-0336  T  +  -0000221T2 
T  being  the  temperature  in  degrees  centigrade  and  D  the  diameter 
of  the  pipe. 


OF 
<  FLOW   THROUGH    PIPES  129 

100.     Critical  velocity  by  the  method  of  colour  bands. 

The  existence  of  the  critical  velocity  has  been  beautifully 
shown  by  Reynolds,  by  the  method  of  colour  bands,  and  his 
experiments  also  explain  why  there  is  a  sudden  change  in  the  law 
connecting  i  and  v. 

"  Water  was  drawn  through  tubes  (Figs.  91  and  92),  out  of 
a  large  glass  tank  in  which  the  tubes  were  immersed,  and  in 
which  the  water  had  been  allowed  to  come  to  rest,  arrangements 
being  made  as  shown  in  the  figure  so  that  a  streak  or  streaks  of 
highly  coloured  water  entered  the  tubes  with  the  clear  water." 


Fig.  91. 


Fig.  92. 

The  results  were  as  follows : — 

"  (1)  When  the  velocities  were  sufficiently  low,  the  streak 
of  colour  extended  in  a  beautiful  straight  line  through  the  tube  " 
(Fig.  91). 

"  (2)  As  the  velocity  was  increased  by  small  stages,  at 
some  point  in  the  tube,  always  at  a  considerable  distance  from  the 
trumpet-shaped  intake,  the  colour  band  would  all  at  once  mix  up 
with  the  surrounding  water,  and  fill  the  rest  of  the  tube  with 
a  mass  of  coloured  water"  (Fig.  92). 

This  sudden  change  takes  place  at  the  critical  velocity. 

That  such  a  change  takes  place  is  also  shown  by  the  apparatus 
illustrated  in  Fig.  88;  when  the  critical  velocity  is  reached  there  is 
a  violent  disturbance  of  the  mercury  in  the  U  tube. 

There  is,  therefore,  a  definite  and  sudden  change  in  the  con- 
dition of  flow.  For  velocities  below  the  critical  velocity,  the  flow 
is  parallel  to  the  tubes,  or  is  "  Stream  Line  "  flow,  but  after  the 
critical  velocity  has  been  passed,  the  motion  parallel  to  the  tube  is 
accompanied  by  eddy  motions,  which  cause  a  definite  change  to 
take  place  in  the  law  of  resistance. 

Barnes  and  Coker*  have  determined  the  critical  velocity  by 
noting  the  sudden  change  of  temperature  of  the  water  when  its 
motion  changes.  They  have  also  found  that  the  critical  velocity, 
as  determined  by  noting  the  velocity  at  which  stream-line  flow 

*  Proceedings  of  the  Royal  Society,  Vol.  LXXIV.  1904;  Phil.  Transactions, 
Royal  Society,  Vol.  xx.  pp.  45—61. 

9 


130  HYDRAULICS 

breaks  up  into  eddies,  is  a  much  more  variable  quantity  than 
that  determined  from  the  points  of  intersection  of  the  two  lines 
as  in  Fig.  90.  In  the  former  case  the  critical  velocity  depends 
upon  the  condition  of  the  water  in  the  tank,  and  when  it  is 
perfectly  at  rest  the  stream  lines  may  be  maintained  at  much 
higher  velocities  than  those  given  by  the  formula  of  Reynolds. 
If  the  water  is  not  perfectly  at  rest,  the  results  obtained  by  both 
methods  agree  with  the  formula. 

Barnes  and  Coker  have  called  the  critical  velocity  obtained  by 
the  method  of  colour  bands  the  upper  limit,  and  that  obtained  by 
the  intersection  of  the  logarithmic  homologues  the  lower  critical 
velocity.  The  first  gives  the  velocity  at  which  water  flowing  from 
rest  in  stream-line  motion  breaks  up  into  eddy  motion,  while  the 
second  gives  the  velocity  at  which  water  that  is  initially  disturbed 
persists  in  flowing  with  eddy  motions  throughout  a  long  pipe,  or 
in  other  words  the  velocity  is  too  high  to  allow  stream  lines  to  be 
formed. 

That  the  motion  of  the  water  in  large  conduits  is  in  a  similar 
condition  of  motion  is  shown  by  the  experiment  of  Mr  Gr.  H. 
Benzenberg*  on  the  discharge  through  a  sewer  12  feet  in  diameter, 
2534  ft.  long. 

In  order  to  measure  the  velocity  of  water  in  the  sewer,  red 
eosine  dissolved  in  water  was  suddenly  injected  into  the  sewer, 
and  the  time  for  the  coloured  water  to  reach  the  outlet  half  a 
mile  away  was  noted.  The  colour  was  readily  perceived  and  it 
was  found  that  it  was  never  distributed  over  a  length  of  more  than 
9  feet.  As  will  be  seen  by  reference  to  section  130,  the  velocities 
of  translation  of  the  particles  on  any  cross  section  at  any  instant 
are  very  different,  and  if  the  motion  were  stream  line  the  colour 
must  have  been  spread  out  over  a  much  greater  length. 

101.  Law  of  frictional  resistance  for  velocities  above  the 
critical  velocity. 

As  seen  from  Reynolds'  formula,  the  critical  velocity  except 
for  very  small  pipes  is  so  very  low  that  it  is  only  necessary  in 
practical  hydraulics  to  consider  the  law  of  frictional  resistance  for 
velocities  above  the  critical  velocity. 

For  any  particular  pipe, 

i  =  Jcvn, 
and  it  remains  to  determine  Jc  and  n. 

From  the   plottings   of   the  results  of  his  own  and   Darcy's 


p.  1173. 


Transactions  Am.S.C.E.  1893;  and  also  Proceedings  Am.S.C.E.,  Vol.  xxvn. 


FLOW   THROUGH   PIPES 


131 


experiments,  Reynolds  found  that  the  law  of  resistance  "  for  all 
pipes  and  all  velocities  "  could  be  expressed  as 

AD3.     /BD    Y 

~p~*  =  V~p~«)     (14)- 


Transposing, 


and 


AP*.D3 
B71     2-M 

«=  T- 


(15), 


A  D3-' 

D  is  diameter  of  pipe,  A  and  B  are  constants,  and  P  is  obtained 
from  formula  (13). 

Taking  the  temperature  in  degrees  centigrade  and  the  metre 
as  unit  length, 

A  -  67,700,000, 

B  -  396, 

-r-*,  J- 


or 

in  which 


-0036T  +  -000221T2' 
"  .  vn  .  P2-"     _  y  .  vn 


67,700,000  D3-'       ] 

7  =  67,700,000 ' 
Values  of  y  when  the  temperature  is  10°  C. 


(16), 


71 

7 

1-75 

0-000265 

1-85 

0-000388 

1-95 

0-000587 

2-00 

0-000704 

The  values  for  A  and  B,  as  given  by  Reynolds,  are,  however, 
only  applicable  to  clean  pipes,  and  later  experiments  show  that 
although 


it  is  doubtful  whether 

p  =  3  -  n, 

as  given  by  Reynolds,  is  correct. 

Value  of  n.  For  smooth  pipes  n  appears  to  be  nearly  175. 
Reynolds  found  the  mean  value  of  n  for  lead  pipes  was  1723.  • 

Saph  and  Schoder*,  in  an  elaborate  series  of  experiments 
carried  out  at  Cornell  University,  have  determined  for  smooth 

*  Transactions  of  the  American  Society  of  Civil  Engineers,  May,  1903.  See 
exercise  31,  page  172. 

9—2 


132  HYDRAULICS 

brass  pipes  a  mean  value  for  n  of  1'75.  Coker  and  Clements 
found  that  n  for  a  brass  pipe  '3779  inches  diameter  was  1*731.  In 
column  5  of  Table  XVI  are  given  values  of  n,  some  taken  from 
Saph  and  Schoder's  paper,  and  others  as  determined  by  the 
author  by  logarithmic  plotting  of  a  large  number  of  experiments. 

It  will  be  seen  that  n  varies  very  considerably  for  pipes  of 
different  materials,  and  depends  upon  the  condition  of  the  surface 
of  a  given  material,  as  is  seen  very  clearly  from  Nos.  3  and  4. 
The  value  for  n  in  No.  3  is  1'72,  while  for  No.  4,  which  is  the 
same  pipe  after  two  years'  service,  the  value  of  n  is  1'93.  The 
internal  surface  had  no  doubt  become  coated  with  a  deposit  of 
some  kind. 

Even  very  small  differences  in  the  condition  of  the  surface, 
such  as  cannot  be  seen  by  the  unaided  eye,  make  a  considerable 
difference  in  the  value  of  n,  as  is  seen  by  reference  to  the  values 
for  galvanised  pipes,  as  given  by  Saph  and  Schoder.  For  large 
pipes  of  riveted  steel,  riveted  wrought  iron,  and  cast  iron,  the 
value  of  n  approximates  to  2. 

The  method,  of  plotting  the  logarithms  of  i  and  v  determined 
by  experiment,  allows  of  experimental  errors  being  corrected 
without  difficulty  and  with  considerable  assurance. 

102.  The  determination  of  the  values  of  C  given  in 
Table  XII. 

The  method  of  logarithmic  plotting  has  been  employed  for 
determining  the  values  of  C  given  in  Table  XII. 

If  values  of  C  are  calculated  by  the  substitution  of  the 
experimental  values  of  v  and  i  in  the  formula 


many  of  the  results  are  apparently  inconsistent  with  each  other 
due  to  experimental  errors. 

The  values  of  C  in  the  table  were,  therefore,  determined  as 
follows. 

Since  i  =  kvn 

and  in  the  Chezy  formula 

v  =  C  v  i»f, 

<--* 


therefore  —  -^  =  kvn 

wO 

and  2  log  C  =  21og  v  -  (log  m  +log  k  +  n  log  v)    ......  (17). 

The  index   n  and   the   coefficient   k   were   determined  for   a 
number  of  cast-iron  pipes. 


FLOW   THROUGH    PIPES  133 

Values  of  C  for  velocities  from  1  to  10  were  calculated.  Curves 
were  then  plotted,  for  different  velocities,  having  C  as  ordinates 
and  diameters  as  abscissae,  and  the  values  given  in  the  table  were 
deduced  from  the  curves. 

The  values  of  C  so  interpolated  differ  very  considerably,  in 
some  cases,  from  the  experimental  values.  The  difficulties 
attending  the  accurate  determination  of  i  and  v  are  very  great, 
and  the  values  of  C,  for  any  given  pipe,  as  calculated  by  substi- 
tuting in  the  Chezy  formula  the  losses  of  head  in  friction  and  the 
velocities  as  determined  in  the  experiments,  were  frequently 
inconsistent  with  each  other. 

As,  for  example,  in  the  pipe  of  3*22  ins.  diameter  given  in 
Table  XVI  which  was  one  of  Darcy's  pipes,  the  variation  of  C  as 
calculated  from  In  and  v  given  by  Darcy  is  from  78'8  to  100. 

On  plotting  log  h  and  log  v  and  correcting  the  readings  so 
that  they  all  lie  on  one  line  and  recalculating  C  the  variation  was 
found  to  be  only  from  95'9  to  101. 

Similar  corrections  have  been  made  in  other  cases. 

The  author  thinks  this  procedure  is  justified  by  the  fact  that 
many  of  the  best  experiments  do  not  show  any  such  inconsistencies. 

An  attempt  to  draw  up  an  interpolated  table  for  riveted  pipes 
was  not  satisfactory.  The  author  has  therefore  in  Table  XIV 
given  the  values  of  C  as  calculated  by  formula  (17),  for  various 
velocities,  and  the  diameters  of  the  pipes  actually  experimented 
upon.  If  curves  are  plotted  from  the  values  of  C  given  in 
Table  XIV,  it  will  be  seen  that,  except  for  low  velocities,  the 
curves  are  not  continuous,  and,  until  further  experimental  evidence 
is  forthcoming  for  riveted  pipes,  the  engineer  must  be  content, 
with  choosing  values  of  C,  which  most  nearly  coincide,  as  far  as 
he  can  judge,  with  the  case  he  is  considering. 

103.  Variation  of  k,  in  the  formula  i  =  kvM,  with  the 
diameter. 

It  has  been  shown  in  section  98  how  the  value  of  k,  for  a 
given  pipe,  can  be  obtained  by  the  logarithmic  plotting  of  i  and  v. 

In  Table  XVI,  are  given  values  of  &,  as  determined  by  the 
author,  by  plotting  the  results  of  different  experiments.  Saph 
and  Schoder  found  that  for  smooth  hard-drawn  brass  pipes 
of  various  sizes  n  varied  between  173  and  1*77,  the  mean  value 
being  175. 

By  plotting  log  d  as  abscissae  and  log  A;  as  ordinates,  as  in 
Fig.  93,  for  these  brass  pipes  the  points  lie  nearly  in  a  straight  line 
which  has  an  inclination  V  with  the  axis  of  d,  such  that 


134 


HYDRAULICS' 


and  the  equation  to  the  line  is,  therefore, 

log  ~k  =  log  y  -  p  log  d, 
where  p  =  1'25, 

and  log  y  =  log  Jc 

when  d  =  I. 

From  the  figure 

y  =  G'000296  per  foot  length  of  pipe. 


i 

•05. 
•04 
•03 

•02 


•01 


•003 


Equation  tu  line 
Log.lo-Log  m,-1-25Lcg  d 

—126 


•02    -03  -04-    -06  -OS  W         -2O 
Log  d 


Fig.  93.    Logarithmic  plottings  of  fe  and  d  to  determine  the  index  p  in  the  formula 

.  _  7  .  vn 

On  the  same  figure  are  plotted  logd  and  log&,  as  deduced 
from  experiments  on  lead  and  glass  pipes  by  various  workers.  It 
will  be  seen  that  all  the  points  lie  very  close  to  the  same  line. 

For  smooth  pipes,  therefore,  and  for  velocities  above  the 
critical  velocity,  the  loss  of  head  due  to  friction  is  given  by 

yvn 

the  mean  value  for  y  being  0'000296,  for  ra,  175,  and  for  p  T25. 

From  which,  v  =  104f  md™, 

or  log  v  =  2-017  +  0-572  log  i  +  0715  log  d. 


FLOW    THROUGH    PIPES  135 

The  value  of  p  in  this  formula  agrees  with  that   given  by 
Reynolds  in  his  formula 


Professor  Unwin*  in  1886,  by  an  examination  of  experiments 
on  cast-iron  pipes,  deduced  the  formula,  for  smooth  cast-iron 
pipes, 


and  for  rough  pipes, 


d 


1-1 


M.  Flamantt  in  1892  examined  carefully  the  experiments 
available  on  flow  in  pipes  and  proposed  the  formula, 

yv^ 

1          ^1-25    ) 

for  all  classes  of  pipes,  and  suggested  for  y  the  following  values : 
Lead  pipes  \ 

G-lass    „  \  -000236  to  '00028, 

Wrought-iron  (smooth)  J 
Cast-iron  new  '000336, 

„       „     in  service  '000417. 

If  the  student  plots  from  Table  XVI,  log  d  as  ordinates,  and 
log  k  as  abscissae,  it  will  be  found,  that  the  points  all  lie  between 
two  straight  lines  the  equations  to  which  are 

log  k  =  log  '00069  -  T25  log  d, 
and  log  Jc  =  log  '00028  - 1'25  log  d. 

Further,  the  points  for  any  class  of  pipes  not  only  lie  between 
these  two  lines,  but  also  lie  about  some  line  nearly  parallel  to 
these  lines.     So  that  p  is  not  very  different  from  1'25. 
From  the  table,  n  is  seen  to  vary  from  1*70  to  2'08. 
A  general  formula  is  thus  obtained, 

7     -00028  to -00069t;r70to2-08Z 
II  =  — -jj^: —  — . 

The  variations  in  y,  nt  and  p  are,  however,  too  great  to  admit 
of  the  formula  being  useful  for  practical  purposes. 
For  new  cast-iron  pipes, 

fe  = -000296  to  -0004  W84*01'9^ 

If  the  pipes  are  lined  with  bitumen  the  smaller  values  of  y  and 
n  may  be  taken. 

*  Industries,  1886. 

t  Annales  des  Fonts  et  Chaussees,  1892,  Vol.  n. 


1.36 


HYDRAULICS 


For  new,  steel,  riveted  pipes, 

•0004to'00054^1'93to2'08Z 


Fig.  94  shows  the  result '  of   plotting  log  k  and  log  d  for  all 
the  pipes  in  Table  XVI  having  a  value  of  n  between  1'92  and  1'94. 
They  are  seen  to  lie  very  close  to  a  line  having  a  slope  of  1'25, 
and  the  ordinate  of  which,  when  d  is  1  foot,  is  '000364. 
•000364^r9:3Z 


Therefore         h  = 


d1 


or  v  = 


very  approximately  expresses  the  law  of  resistance  for  particular 
pipes  of  wood,  new  cast  iron,  cleaned  cast  iron,  and  galvanised 
iron. 


and  lo    d  rronv  Table  16, 


Loyk. 


plotting  s  of  loq  h 
og  c 

to  deter\nin 
formula^  i,  = 


he  indea>  p  in-  the. 

is  atxwbl'93 


Fig.  94. 

Taking  a  pipe  1  foot  diameter  and  the  velocity  as  3  feet  per 
second,  the  value  of  i  obtained  by  this  formula  agrees  with  that 
from  Darcy's  formula  for  clear  cast-iron  pipes  within  1  per  cent. 

Use  of  the  logarithmic  formula  for  practical  calculations.  A 
very  serious  difficulty  arises  in  the  use  of  the  logarithmic 
formula,  as  to  what  value  to  give  to  n  for  any  given  case,  and 
consequently  it  has  for  practical  purposes  very  little  advantage 
over  the  older  and  simpler  formula  of  Chezy. 


TABLE  XVI. 


Experimenter 

Kind  of  pipe 

Diameter 
(in  ins.) 

Velocity  in 
ft.  per  sec. 
from          to 

Value  of  n 
in  formula 

i  =  kvn 

Value  of  k 
in  formula 

/  =  kv» 

Noble 

Wood 

44 

3-46  _  4-415 

1-73 

•0001254 

J} 

55 

54 

2-28  —  4-68 

1-75 

•000083 

Marx,  Wing  ) 

55 

72-5 

1         -  4 

1-72 

•000061 

and  Hoskins  } 

55 

72-5 

1        —  5-5 

1-93 

•000048 

raltner  Kitcham 

Riveted 

3 

1-88 

•00245 

H.  Smith 

Wrought 

11 

1-81 

•000515 

55 

iron  or  steel 

11| 

1-90 

•000470 

H 

55 

15 

1-94 

•000270 

Kinchling 

55 

38 

•505—  1-254 

2-0 

•000099 

Herschel 

42 

2-10  —  4-99 

1-93 

•00011 

55 

55 

48 

2          -  5  (?) 

2-0 

•000090 

Marx,  Wing  ) 

55 

72 

1        —  4 

1-99 

•000055 

and  Hoskins  ( 

55 

72 

1        —  5-5 

1-85 

•000077 

Herschel 

55 

103 

1         -  4-5 

2-08 

•000036 

Darcy 

Cast  iron 

3-22 

•289—10-71 

1-97 

•00156 

59 

new 

5-39 

•48  —15-3 

1-97 

•00079 

,9 

55 

7-44 

•673—16-17 

1-956 

•00062 

99 

55 

12 

1-779 

•000323 

Williams 

55 

16-25 

1-858 

•000214 

Lampe 

55 

16-5 

2-48  —  3-09 

1-80 

•000267 

57 

55 

19-68 

1-38  —  3-7 

1-84 

•00022 

Sherman 

55 

36 

4         -  7 

2* 

•000062 

Stearns 

55 

48 

1-243—  3-23 

1-92 

•0000567 

(tubbell&Fenkell 

55 

30 

2 

•00003 

Darcy 

Cast  iron 

1-4136 

•167—  2-077 

1-99 

•0098 

old  and 

3-1296 

•403—  3-747 

1-94 

•0035 

55 

tuberculated 

9-575 

1-007—12-58 

1-98 

•0009 

Sherman 

20 

2-71  —  5-11 

ii 

36 

1-1     —  4-5 

2 

•000105 

Fitzgerald 

55 

48 

1-176—  3-533 

2-04 

•000083 

n 

55 

48 

1-135—  3-412 

2-00 

•000085 

Darcy 

Cast-iron 

1-4328 

•371—  3-69 

1-85 

•0041 

old  pipes 

3-1536 

•633—  5-0 

1-97 

•00185 

cleaned 

11-68 

•8     —10-368 

2-0 

•000375 

Fitzgerald 

55 

55 
55 

48 
48 

3-67  —  5-6 
•395—  7-245 

2-02 
1-94 

•000082 
•000059 

Darcy 

Sheet  -iron 

1-055 

-098—  8-225 

1-76 

•0074 

j5 

3-24 

•328—12-78 

1-81 

•00154 

7-72 

•591—19-72 

1-78 

•00059 

>5 

H 

11-2 

1-296—10-52 

1-81 

•00039 

Gas 

•48 

•113—  3-92 

1-83 

•0278 

pi 

1-55 

•205—  8-521 

1-86 

•00418 

„ 

55 

1-91 

•0072 

aph  and  Schoder 

Galvanised 

•364 

1-96 

•0352 

? 

. 

•494 

1-91 

•0181 

•623 

1-86 

•0132 

,9 

9, 

•824 

1-80 

•0095 

55 

55 

1-048 

1-93 

•0082 

tt 

Hard-drawn 
brass 

15  pipes 
up  to  1-84 

1-75 

•00025  to 
•00035 

Reynolds 

Lead 

1-732 

Darcy 

•55 

1-761 

•0126 

•/ 

55 

1-61 

1-783 

•00425 

1.38 


HYDRAULICS 


TABLE  XVII. 

Showing  reasonable  values  of  y,  and  n,  for  pipes  of  various 
kinds,  in  the  formula, 

L    YV*I 


Reasonable 

values  for 

7 

n 

7 

n 

Clean  cast-iron  pipes 

•00029  to  -000418 

1-80  to  1-97 

•00036 

1-93 

Old  cast-iron  pipes 

•00047  to  -00069 

1-94  to  2-04 

•00060 

2 

Riveted  pipes 
Galvanised  pipes 

•00040  to  -00054 
•00035  to  -00045 

1-93  to  2-08 
1-80  to  1-96 

•00050 
•00040 

2 

1-88 

Sheet-iron  pipes  cover- 
ed with  bitumen 

•00030  to  -00038 

1-76  to  1-81 

•00034 

1-78 

Clean  wood  pipes 
Brass  and  lead  pipes 

•00056  to  -00063 

1-72  to  1-75 

•00060 
•00030 

1-75 
1-75 

When  further  experiments  have  been  performed  on  pipes,  of 
which  the  state  of  the  internal  surfaces  is  accurately  known,  and 
special  care  taken  to  ensure  that  all  the  loss  of  head  in  a  given 
length  of  pipe  is  due  to  friction  only,  more  definiteness  may  be 
given  to  the  values  of  y,  ?i,  and  p. 

Until  such  evidence  is  forthcoming  the  simple  Chezy  formula 
may  be  used  with  almost  as  much  confidence  as  the  more 
complicated  logarithmic  formula,  the  values  of  C  or  /  being  taken 
from  Tables  XII — XIY.  Or  the  formula  h  =  kvn  may  be  used, 
values  of  k  and  n  being  taken  from  Table  XYI,  which  most  nearly 
fits  the  case  for  which  the  calculations  are  to  be  made. 

104.     Criticism  of  experiments. 

The  difficulty  of  differentiating  the  loss  of  head  due  to  friction 
from  other  sources  of  loss,  such  as  loss  due  to  changes  in  direction, 
change  in  the  diameter  of  the  pipe  and  other  causes,  as  well  as  the 
possibilities  of  error  in  experiments  on  long  pipes  of  large  diameter, 
makes  many  experiments  that  have  been  performed  of  very  little 
value,  and  considerably  increases  the  difficulty  of  arriving  at 
correct  formulae. 

The  author  has  found  in  many  cases,  when  log  i  and  log  d  were 
plotted,  from  the  records  of  experiments,  that,  although  the  results 
seemed  consistent  amongst  themselves,  yet  compared  with  other 
experiments,  they  seemed  of  little  value. 


FLOW   THROUGH    PIPES 


139 


The  value  of  n  for  one  of  Couplet's*  experiments  on  a  lead  and 
earthenware  pipe  being  as  low  as  1*56,  while  the  results  of  an 
experiment  by  Simpson  t  on  a  cast-iron  pipe  gave  n  as  2'5.  In  the 
latter  case  there  were  a  number  of  bends  in  the  pipe. 

In  making  experiments  for  loss  of  head  due  to  friction,  it  is 
desirable  that  the  pipe  should  be  of  uniform  diameter  and  as 
straight  as  possible  between  the  points  at  which  the  pressure  head 
is  measured.  Further,  special  care  should  be  taken  to  ensure  the 
removal  of  all  air,  and  that  a  perfectly  steady  flow  is  established 
at  the  point  where  the  pressure  is  taken. 

105.     Piezometer  fittings. 

It  is  of  supreme  importance  that  the 
piezometer  connections  shall  be  made 
so  that  the  difference  in  the  pressures 
registered  at  any  two  points  shall  be 
that  lost  by  friction,  and  friction  only, 
between  the  points. 

This  necessitates  that  there  shall 
be  no  obstructions  to  interfere  with  the 
free  flow  of  the  water,  and  it  is,  there- 
fore, very  essential  that  all  burrs  shall 
be  removed  from  the  inside  of  the  pipe. 

In  experiments  on  small  pipes  in 
the  laboratory  the  best  results  are  no 
doubt  obtained  by  cutting  the  pipe 
completely  through  at  the  connection 
as  shown  in  Fig.  95,  which  illustrates 
the  form  of  connection  used  by  Dr 
Coker  in  the  experiments  cited  on 

page  129.     The  two  ends  of  the  pipe  are  not  more  than 
of  an  inch  apart. 

Fig.  96  shows  the  method  adopted  by  Marx,  Wing  and  Hoskins 
in  their  experiments  on  a  72-inch  wooden  pipe  to  ensure  a  correct 
reading  of  the  pressure. 

The  gauge  X  was  connected  to  the  top  of  the  pipe  only  while 
Y  was  connected  at  four  points  as  shown. 

Small  differences  were  observed  in  the  readings  of  the  two 
gauges,  which  they  thought  were  due  to  some  accidental  circum- 
stance affecting  the  gauge  X  only,  as  no  change  was  observed 
in  the  reading  of  Y  when  the  points  of  communication  to  Y  were 
changed  by  means  of  the  cocks. 

*  Hydraulics,  Hamilton  Smith,  Junr. 

t  Proceedings  of  the  Institute  of  Civil  Engineers,  1855. 


Fig.  95. 


140 


HYDRAULICS 


106.     Effect  of  temperature  on  the  velocity  of  flow. 

Poiseuille  found  that  by  raising  the  temperature  of  the  water 
from  50°  C.  to  100°  C.  the  discharge  of  capillary  tubes  was 
doubled. 


Fig.  96.     Piezometer  connections  to  a  wooden  pipe. 

Reynolds*  showed  that  for  pipes  of  larger  diameter,  the  effect 
of  changes  of  the  temperature  was  very  marked  for  velocities 
below  the  critical  velocity,  but  for  velocities  above  the  critical 
velocity  the  effect  is  comparatively  small. 

The  reason  for  this  is  seen,  at  once,  from  an  examination  of 
Reynolds'*  formula.  Above  the  critical  velocity  n  does  not  differ 
very  much  from  2,  so  that  P2~u  is  a  small  quantity  compared  with 
its  value  when  n  is  1. 

Saph  and  Schodert,  for  velocities  above  the  critical  velocity, 
found  that,  as  the  temperature  rises,  the  loss  of  head  due  to 
friction  decreases,  but  only  in  a  small  degree.  For  brass  pipes  of 
small  diameter,  the  correction  at  60°  F.  was  about  4  per  cent,  per 

*  Scientific  Papers,  Vol.  n. 

t  See  also  Barnes  and  Coker,  Proceedings  of  the  Eoyal  Society,  Vol.  LXX.  1904  ; 
Coker  and  Clements,  Transactions  of  the  Royal  Society,  Vol.  cci.  Proceedings 
Am.S.G.E.  Vol.  xxix. 


FLOW   THROUGH   PIPES  141 

10  degrees  F.     With  galvanised  pipes  the  correction  appears  to 
be  from  1  per  cent,  to  5  per  cent,  per  10  degrees  F. 

Since  the  head  lost  increases,  as  the  temperature  falls,  the 
discharge  for  any  given  head  diminishes  with  the  temperature, 
but  for  practical  purposes  the  correction  is  generally  negligible. 

107.     Loss  of  head  due  to  bends  and  elbows. 

The  loss  of  head  due  to  bends  and  elbows  in  a  long  pipe  is 
generally  so  small  compared  with  the  loss  of  head  due  to  friction 
in  the  straight  part  of  the  pipe,  that  it  can  be  neglected,  and 
consequently  the  experimental  determination  of  this  quantity  has 
not  received  much  attention. 

Weisbach*,  from  experiments  on  a  pipe  1J  inches  diameter, 
with  bends  of  various  radii,  expressed  the  loss  of  head  as 


r  being  the  radius  of  the  pipe,  ~R  the  radius  of  the  bend  on  the 
centre  line  of  the  pipe  and  v  the  velocity  of  the  water  in  feet  per 
second.  If  the  formula  be  written  in  the  form 

'2 


at' 


/\+ 

the  table  shows  the  values  of  a  for  different  values  of  ^  . 

XV 

r 
B 

•1  -157 

•2  -250 

•5  -526 

St  Venantt  has  given  as  the  loss  of  head  h^  at  a  bend, 
AB  =  '00152  ^  y^M)-l|Ay|  nearly, 

I  being  the  length  of  the  bend  measured  on  the  centre  line  of  the 
bend  and  d  the  diameter  of  the  pipe. 
When  the  bend  is  a  right  angle 


l_     /~d_v      /d 
R  V  R  ~  2  V  R ' 


When  5-1,  '5,  '2, 

n 


- 1-57,          Ml,  702 

and  7*B=   "157  „-,     'Ills-,     '07^-. 


*  Mechanics  of  Engineering. 
t  Comptes  Rendus,  1862. 


142 


HYDRAULICS 


Recent  experiments  by  Williams,  Hubbell  and  Fenkell  *  on  cast- 
iron  pipes  asphalted,  by  Saph  and  Schoder  on  brass  pipes,  and 
others  by  Alexander  t  on  wooden  pipes,  show  that  the  loss  of  head 
in  bends,  as  in  a  straight  pipe,  can  be  expressed  as 

/IB  =  kvn, 
n  being  a  variable  for  different  kinds  of  pipes,  while 


d* 

y  being  a  constant  coefficient  for  any  pipe. 

For  the  cast-iron  pipes  of  Hubbell  and  Fenkell,  y,  n,  m,  and  p 
have  approximately  the  following  values. 


Diameter  of  pipe 

7 

m 

n 

f 

12" 

•0040 

0-83 

1-78 

1-09 

16" 

?5 

5? 

1-86 

?? 

30" 

5? 

?5 

2-0 

J5 

» 

When  v  is  3  feet  per  second  and  ^-  is  ;},  the  bend  being  a  right 

angle,  the  loss  of  head   as  calculated  by  this  formula  for  the 

,      .       .    -206&;2 
12-inch  pipe  is  —  ~  - 


,      ,,      Qn  •     -,      • 

*or  the  ^0-lncn  P1P&   ~^~  —  • 


For  the  brass  pipes  of  Saph  and  Schoder,  2  inches  diameter, 
Alexander  found 


7iB  =  '00858   5 

/T« 

and  for  varnished  wood  pipes  when     -  is  less  than  0'2, 
7iB  =  '008268   i 


and  when  ^  is  between  0*2  and  0'5, 


He  further  found  for  varnished  wood  pipes  that,  a  bend  of 
radius  equal  to  5  times  the  radius  of  the  pipe  gives  the  minimum 
loss  of  head,  and  that  its  resistance  is  equal  to  a  straight  pipe  3'38 
times  the  length  of  the  bend. 

Messrs  Williams,  Hubbell  and  Fenkell  also  state  at  the  end  of 
their  elaborate  paper,  that  a  bend  having  a  radius  equal  to  2J 

*  Proc.  Amer.  Soc.  Civil  Engineers,  Vol.  xxvu. 
t  Proc.  Inst>  Civil  Engineers,  Vol.  CLIX. 


FLOW  THROUGH   PIPES  143 

diameters,  offers  less  resistance  to  the  flow  of  water  than  those  of 
longer  radius.  It  should  not  be  overlooked,  however,  that  although 
the  loss  of  head  in  a  bend  of  radius  equal  to  2J  diameters  of  the 
pipe  is  less  than  for  any  other,  it  does  not  follow  that  the  loss  of 
head  per  unit  length  of  the  pipe  measured  along  its  centre  line 
has  its  minimum  value  for  bends  of  this  radius. 

108.  Variations  of  the  velocity  at  the  cross  section  of  a 
cylindrical  pipe. 

Experiments  show  that  when  water  flows  through  conduits  of 
any  form,  the  velocities  are  not  the  same  at  all  points  of  any 
transverse  section,  but  decrease  from  the  centre  towards  the 
circumference. 

The  first  experiments  to  determine  the  law  of  the  variation  of 
the  velocity  in  cylindrical  pipes  were  those  of  Darcy,  the  pipes 
varying  in  diameter  from  7*8  inches  to  19  inches.  A  complete 
account  of  the  experiments  is  to  be  found  in  his  Recherches 
Experimentales  dans  les  tuyaux. 

The  velocity  was  measured  by  means  of  a  Pitot  tube  at  five 
points  on  a  vertical  diameter,  and    KVV  ,xxxxxxxxxxxxxx^^^^^ 
the    results    plotted   as    shown  in 
Fig.  97. 

Calling  V  the  velocity  at  the 
centre  of  a  pipe  of  radius  R,  u  the 
velocity  at  the  circumference,  vm 
the  mean  velocity,  v  the  velocity 
at  any  distance  r  from  the  centre, 


and  i  the   loss   of   head  per   unit 

length  of  the  pipe,  Darcy  deduced  the  formulae 


&     ^T     47    /^r 

and  vm  =  -  =  -  =\-~k  v  R^. 

When  the  unit  is  the  metre  the  value  of  k  is  11*3,  and  20*4  when 
the  unit  is  the  English  foot. 

Later  experiments  commenced  by  Darcy  and  continued  by 
Bazin,  on  the  distribution  of  velocity  in  a  semicircular  channel, 
the  surface  of  the  water  being  maintained  at  the  horizontal 
diameter,  and  in  which  it  was  assumed  the  conditions  were  similar 
to  those  in  a  cylindrical  pipe,  showed  that  the  velocity  near  the 
surface  of  the  pipe  diminished  much  more  rapidly  than  indicated 
by  the  formula  of  Darcy. 


144  HYDRAULICS 

Bazin  substituted  therefore  a  new  formula, 

CD, 

/"i 

or  since  vm  =  C  *Jmi  =  —^  J^i 

V-t;  =  53^  /_r  y  (2) 

It  was  open  to  question,  however,  whether  the  conditions  of  flow 
in  a  semicircular  pipe  are  similar  to  those  in  a  pipe  discharging 
full  bore,  and  Bazin  consequently  carried  out  at  Dijon*,  experi- 
ments on  the  distribution  of  velocity  in  a  cement  pipe,  2*73  feet 
diameter,  the  discharge  through  which  was  measured  by  means 
of  a  weir,  and  the  velocities  at  different  points  in  the  transverse 
section  by  means  of  a  Pitot  tubet. 

From  these  experiments  Bazin  concluded  that  both  formulae  (1) 
and  (2)  were  incorrect  and  deduced  the  three  formulae 


•5{l-x/l--95(£)2} (5), 


V-tJ  =  Vl*53'5|l 

the  constants  in  these  formulae  being  obtained  from  Bazin's  by 
changing  the  unit  from  1  metre  to  the  English  foot. 

Equation  (5)  is  the  equation  to  an  ellipse  to  which  the  sides  of 
the  pipes  are  not  tangents  but  are  nearly  so,  and  this  formula 
gives  values  of  v  near  to  the  surface  of  the  pipe,  which  agree  much 
more  nearly  with  the  experimental  values,  than  those  given  by 
any  of  the  other  formulae. 

Experiments  of  Williams,  HuHbdl  and  Fenkell+.  An  elaborate 
series  of  experiments  by  these  three  workers  have  been  carried  out 
to  determine  the  distribution  of  velocity  in  pipes  of  various 
diameters,  Pitot  tubes  being  used  to  determine  the  velocities. 

The  pipes  at  Detroit  were  of  cast  iron  and  had  diameters  of  12, 
16,  30  and  42  inches  respectively. 

The  Pitot  tubes  §  were  calibrated  by  preliminary  experiments 
on  the  flow  through  brass  tubes  2  inches  diameter,  the  total 

*  "  Memoire  de  1'Academie  des  Sciences  de  Paris,  Kecueil  des  Savants  Etrangeres," 
Vol.  xxxn.  1897.  Proc.  Am.S.C.E.  Vol.  xxvu.  p.  1042. 

t  See  page  241. 

£  "Experiments  at  Detroit,  Mich.,  on  the  effect  of  curvature  on  the  flow  of 
water  in  pipes,"  Proc.  Am.S.C.E.  Vol.  xxvn.  p.  313. 

§  See  page  246. 


FLOW  THROUGH   PIPES  145 

discharge  being  determined  by  weighing,  and  the  mean  velocity 
thus  determined.  From  the  results  of  their  experiments  they 
came  to  the  conclusion  that  the  curve  of  velocities  should  be  an 
ellipse  to  which  the  sides  of  the  pipe  are  tangents,  and  that  the 
velocity  at  the  centre  of  the  pipe  V  is  l'19vm,  vm  being  the  mean 
velocity. 

These  results  are  consistent  with  those  of  Bazin.     His  experi- 

y 
mental  value  for  —  for  the  cement  pipe  was  1'1675,  and  if  the 

vm 

constant  '95,  in  formula  (5),  be  made  equal  to  1,  the  velocity  curve 
becomes  an  ellipse  to  which  the  walls  of  the  pipe  are  tangents. 

The  ratio  —  can  be  determined  from  any  of  Bazin's  formulae. 


Substituting       p  for  \R    in  (1),  (3),  (4)  or  (5),  the  value  of 
v  at  radius  r  can  be  expressed  by  any  one  of  them  as 


i 


c 

Then,  since  the  flow  past  any  section  in  unit  time  is  -Um^R2,  and 
that  the  flow  is  also  equal  to 

fE 
0 

therefore  ^m7rR2  =  2?r  I    |Y—  ~7y^/(~|?  }\Tdr (6). 

Substituting  for  /(-5J ,  its  value  -^3-  from  equation  (1),  and 
integrating, 

^  =  1  +  —^    ...(7), 


(T  \ 
^}  from  equation  (4), 


V     ,23 

~=    H 


so  that  the  ratio  —  is  not  very  different  when  deduced  from  the 

vm 

simple  formula  (2)  or  the  more  complicated  formula  (4). 
When  C  has  the  values 

0  =  80,       100,   120, 

from  (8)  —  =  1-287,  T23,  T19. 

Vm 

The  value  of  C,  in  the  30-inch  pipe  referred  to  above,  varied 
between  109'6  and  123'4  for  different  lengths  of  the  pipe,  and 
L.  H.  10 


<: V— 

< V- — 


-*— _JB 


146  HYDRAULICS 

the  mean  value  was  116,  so  that  there  is  a  remarkable  agreement 
between  the  results  of  Bazin,  and  Williams,  Hubbell  and  Fenkell. 

The  velocity  at   the  surface  of  a  pipe.     Assuming  that   the 
velocity  curve  is  an  ellipse  to  which 
the  sides  of  the  pipe  are  tangents,  as     _ 
in  Fig.  98,  and  that  Y  =  l'19vm,  the 
velocity  at  the  surface  of  the  pipe 
can  readily  be  determined. 

Let  u  =  the  velocity  at  the  surface 
of  the  pipe  and  v  the  velocity  at  any     •• 
radius  r. 

Let  the  equation  to  the  ellipse  be  FlS-  98< 

^2  ~2 

—  +    --1 

B*    P 

in  which  x  =  v-u, 

and  b  =  V  -  u. 

Then,  if  the  semi-ellipse  be  revolved  about  its  horizontal  axis, 
the  volume  swept  out  by  it  will  be  firR26,  and  the  volume  of 
discharge  per  second  will  be 

Zirrdr .v  =  TrW.u  +  f  7rR2£, 

•'•  vm  =  u  +  |(Y  -  u)  =  lu  +  §  x  l'19vm, 
and  u  =  '621vm. 

Using  Bazin' s  elliptical  formula,  the  values  of  —  for 
C  =  80,      100,     120, 

are  --'552,  '642,    '702. 

vm 

The  velocities,  as  above  determined,  give  the  velocity  of 
translation  in  a  direction  parallel  to  the  pipe,  but  as  shown  by 
Reynolds'  experiments  the  particles  of  water  may  have  a  much 
more  complicated  motion  than  here  assumed. 

109.  Head  necessary  to  give  the  mean  velocity  vm  to 
the  water  in  the  pipe. 

It  is  generally  assumed  that  the  head  necessary  to  give  a  .mean 

Vm 

velocity  vm  to  the  water  flowing  in  a  pipe  is  7^-,  which  would  be 

correct  if  all  the  particles  of  water  had  a  common  velocity  vm . 

If,  however,  the  form  of  the  velocity  curve  is  known,  and  on  the 
assumption  that  the  water  is  moving  in  stream  lines  with  definite 
velocities  parallel  to  the  axis  of  the  pipe,  the  actual  head  can 
be  determined  by  calculating  the  mean  kinetic  energy  per  Ib.  of 

water  flowing  in  the  pipe,  and  this  is  slightly  greater  than  -- . 


FLOW  THROUGH   PIPES  147 

As  before,  let  v  be  the  velocity  at  radius  r. 
The  kinetic  energy  of  the  quantity  of  water  which  flows  past 
any  section  per  second 

w  .  Z-n-rdr  .  v  .  ^-  , 
o  2gr' 

w  being  the  weight  of  1  c.  ft.  of  water. 
The  kinetic  energy  per  lb.,  therefore, 


=  / 

J 


_  J  o 


E 

w . 
o 


v/2 


•I'm 


f(S>] 


rdr 


.(9). 


The  simplest  value  for  /  (T>)  is  that  of  Bazin's  formula  (1) 
above,  from  which 

v=#, 


and  f  (  £  )  =  38 


Substituting  these  values  and  integrating,  the  kinetic  energy 
per  lb.  is  ^— ,  and  when 

C  is  80,     100, 
a  is  112,  1-076. 

On  the  assumption  that  the  velocity  curve  is  an  ellipse  to  which 
the  walls  of  the  pipe  are  tangents  the  integration  is  easy,  and  the 
value  of  a  is  1*047. 

Using  the  other  formulae  of  Bazin  the  calculations  are  tedious 
and  the  values  obtained  differ  but  slightly  from  those  given. 

The  head  necessary  to  give  a  mean  velocity  vm  to  the  water  in 

the  pipe  may  therefore  be  taken  to  be  ~- ,  the  value  of  a  being 

about  1*12.  This  value*  agrees  with  the  value  of  1*12  for  a, 
obtained  by  M.  Boussinesq,  and  with  that  of  M.  J.  Delemer  who 
finds  for  a  the  value  1*1346. 

110.    Practical  problems. 

Before  proceeding  to  show  how  the  formulae  relating  to  the 
loss  of  head  in  pipes  may  be  used  for  the  solution  of  various 
problems,  it  will  be  convenient  to  tabulate  them. 
*  Flamant's  Hydraulique. 

10—2 


148  HYDRAULICS 

NOTATION. 
h  =  loss  of  head  due  to  friction  in  a  length  I  of  a  straight  pipe. 

i  =  the  virtual  slope  =  y . 

v  =  the  mean  velocity  of  flow  in  the  pipe. 
d  =  the  diameter. 
m  =  the  hydraulic  mean  depth 

=  — —       ^ea. —  =  —  =  T  when  the  pipe  is  cylindrical  and  full. 

Wetted  Perimeter     P     4 

,      vH      4v2l 

Formula  1.  h  =  7^—  =  7^-, . 

Cm     C  d 

7i       v2 
This  may  be  written    y  =  ™—  , 

or  v  —  C  vmi. 

The  values  of  C  for  cast-iron  and  steel  pipes  are  shown  in 
Tables  XII  and  XIV. 

4fZ^2 
Formula  2.  h  =  ^- — -3 , 

*-  in  this  formula  being  equal  to  7^2  of  formula  (1). 
"9 

Values  of  f  are  shown  in  Table  XIII. 

Either  of  these  formulae  can  conveniently  be  used  for 
calculating  h,  v,  or  d  when  /,  and  Z,  and  any  two  of  three 
quantities  /&,  v,  and  d,  are  known. 

Formula  3.  As  values  of  C  and  /  cannot  be  remembered  for 
variable  velocities  and  diameters,  the  formulae  of  Darcy  are 
convenient  as  giving  results,  in  many  cases,  with  sufficient 
accuracy.  For  smooth  clean  cast-iron  pipes 


For  rough  and  dirty  pipes 


mi. 


FLOW   THROUGH   PIPES  149 

If  d  is  the  unknown,  Darcy's  formulae  can  only  be  used  to  solve 

for  d  by  approximation.    The  coefficient  (  1  +  ^75-7  )  is  first  neglected 

\       LZa/ 

and  an  approximate  value  of  d  determined.  The  coefficient  can 
then  be  obtained  from  this  approximate  value  of  d  with  a  greater 
degree  of  accuracy,  and  a  new  value  of  d  can  then  be  found,  and 
so  on.  (See  examples.) 

Formula  4.     Known  as  the  logarithmic  formula. 

r_* 


or 


d*  ' 

h     .     y  .  vn 
- 


Values  of  y,  n,  and  p  are  given  on  page  138. 
By  taking  logarithms 

log  h  =  log  y  +  n  log  v  +  log  I  —  p  log  d, 
from  which  h  can  be  found  if  I,  v,  and  d  are  known. 
If  h,  I,  and  d  are  known,  by  writing  the  formula  as 

n  log  v  =  log  Tz,  —  log  I  -  log  y  +  p  log  d, 
-u  can  be  found. 

If  h,  I,  and  v  are  known,  d  can  be  obtained  from 
p  log  d  =  log  y  +  n  log  v  +  log  I  -  log  /i. 

This  formula  is  a  little  more  cumbersome  to  use  than  either  (1)  or 
(2)  but  it  has  the  advantage  that  y  is  constant  for  all  velocities. 
Formula  5.     The  head  necessary  to  give  a  mean  velocity  v  to 

l"12i;2 
the  water  flowing  along  the  pipe  is  about  —  ^  —  ,  but  it  is  generally 

v2 
convenient  and  sufficiently  accurate  to  take  this  head  as  g-,  as 

was  done  in  Fig.  87.     Unless  the  pipe  is  short  this  quantity  is 
negligible  compared  with  the  friction  head. 

Formula  6.     The  loss  of  head  at  the  sharp-edged  entrance  to  a 

•K-Jl 

pipe  is  about  -~—-  and  is  generally  negligible. 

Formula  7.  The  loss  of  head  due  to  a  sudden  enlargement  in 
a  pipe  where  the  velocity  changes  from  Vi  to  v2  is  ^  2  • 

Formula  8.  The  loss  of  head  at  bends  and  elbows  is  a  very 
variable  quantity.  It  can  be  expressed  as  equal  to  -^-  in  which 
a  varies  from  a  very  small  quantity  to  unity. 

Problem  1.  The  difference  in  level  of  the  water  in  two  reservoirs  is  h  feet, 
Fig.  99,  and  they  are  connected  by  means  of  a  straight  pipe  of  length  I  and 
diameter  d;  to  find  the  discharge  through  the  pipe. 


150 


HYDRAULICS 


Let  Q  be  the  number  of  cubic  feet  discharged  per  second.  The  head  h  is  utilised 
in  giving  velocity  to  the  water  and  in  overcoming  resistance  at  the  entrance  to  the 
pipe  and  the  frictional  resistances. 


Fig.  99.    Pipe  connecting  two  reservoirs. 

Let  v  be  the  mean  velocity  of  the  water.     The  head  necessary  to  give  the  water 
this  mean  velocity  may  be  taken  as  —    — ,  and  to  overcome  the  resistance  at  the 


entrances 
Then 


l-12t;2      -5t?2 
~~  ^ 


2g.  d 
Or  using  in  the  expression  for  friction,  the  coefficient  C, 

h  =  -0174tr  +  -0078v2  +  ~ 


If  -  is  greater  than  300  the  head  lost  due  to  friction  is  generally  great  compared 

a 
with  the  other  quantities,  and  these  may  be  neglected. 

Then  h= 


and 


2gd    _C2 
C      /dh 

=2VT- 


As  the  velocity  is  not  known,  the  coefficient  C  cannot  be  obtained  from  the 
table,  but  an  approximate  value  can  be  assumed,  or  Darcy's  value 


0  =  394 


J 


0  =  278 


I2d  +  l 


I2d  +  l 


for  clean  pipes, 
if  the  pipe  is  dirty, 


and 

can  be  taken. 

An  approximation  to  v — which  in  many  cases  will  be  sufficiently  near  or  will  be 
as  near  probably  as  the  coefficient  can  be  known — is  thus  obtained.  From  the 
table  a  value  of  C  for  this  velocity  can  be  taken  and  a  nearer  approximation  to 
v  determined. 

Then  Q=^d*.v. 

yvnl 
The  velocity  can  be  deduced  directly  from  the  logarithmic  formula  h=  -=^  > 

provided  7  and  n  are  known  for  the  pipe. 


FLOW   THROUGH   PIPES  151 

The  hydraulic  gradient  is  EF. 

At  any  point  C  distant  x  from  A  the  pressure  head  —  is  equal  to  the  distance 
between  the  centre  of  the  pipe  and  the  hydraulic  gradient.  The  pressure  head 
just  inside  the  end  A  of  the  pipe  is  ft  A  --  5  —  ,  and  at  the  end  B  the  pressure  head 
must  be  equal  to  h#.  The  head  lost  due  to  friction  is  h,  which,  neglecting  the 
small  quantity  —  ~  ,  is  equal  to  the  difference  of  level  of  the  water  in  the  two 

tanks. 

Example  1.  A  pipe  3  inches  diameter  200  ft.  long  connects  two  tanks,  the 
difference  of  level  of  the  water  in  which  is  10  feet,  and  the  pressure  is  atmospheric. 
Find  the  discharge  assuming  the  pipe  dirty. 


Using  Darcy's  coefficient 


=  3-88  ft.  per  sec. 

For  a  pipe  3  inches  diameter,  and  this  velocity,  C  from  the  table  is  about  69,  so 
that  the  approximation  is  sufficiently  near. 

•00064t?i*».  I 
Taking  h  = -^ , 

i>  =  3-88  ft.  per  sec., 

gives  v  =  3 -85  ft.  per  sec. 

Example  2.  A  pipe  18  inches  diameter  brings  water  from  a  reservoir  100  feet 
above  datum.  The  total  length  of  the  pipe  is  15,000  feet  and  the  last  5000  feet 
are  at  the  datum  level.  For  this  5000  feet  the  water  is  drawn  off  by  service  pipes  at 
the  uniform  rate  of  20  cubic  feet  per  minute,  per  500  feet  length.  Find  the  pressure 
at  the  end  of  the  pipe. 

The  total  quantity  of  flow  per  minute  is 

5000  x  20     4 

Q  = =  200  cubic  feet  per  minute. 

ouo 

Area  of  the  pipe  is  1-767  sq.  feet. 

The  velocity  in  the  first  10,000  feet  is,  therefore, 

„=_??!!_.=  1-888  ft.  per  sec. 

The  head  lost  due  to  friction  in  this  length,  is 

k_*-f-  10,000.1-8882 

In  the  last  5000  feet  of  the  pipe  the  velocity  varies  uniformly.  At  a  distance 
x  feet  from  the  end  of  the  pipe  the  velocity  is  . 

In  a  length  dx  the  head  lost  due  to  friction  is 

4./t  l-888a.a?ads 
:    20.1-5.5000*    ' 
and  the  total  loss  by  friction  is 

4/.  1-8882       pooo         _4/.  (1-888)2  5000 
"o-20.1-5.50002J0  20.1-5     '    3 

The  total  head  lost  due  to  friction  in  the  whole  pipe  is,  therefore, 

H=     4{  ,. 


152  HYDRAULICS 

Taking  /  as  -0082,  H  =  14-3  feet. 

Neglecting  the  velocity  head  and  the  loss  of  head  at  entrance,  the  pressure  head 
at  the  end  of  the  pipe  is  (100  -  H)  feet  =  85-7  feet. 

Problem  2.     Diameter  of  pipe  to  give  a  given  discharge. 

Eequired  the  diameter  of  a  pipe  of  length  I  feet  which  will  discharge  Q  cubic  feet 
per  second  between  the  two  reservoirs  of  the  last  problem. 
Let  v  be  the  mean  velocity  and  d  the  diameter  of  the  pipe. 

Then  v=-Q-    (1), 


and 

Therefore, 

Squaring  and  transposing, 
c 
If  I  is  long  compared  with  d, 

and 


0-0406.  Q2d_  6-5ZQ2 

h  &h    {  >' 


Q  /dh 

r-=cv  47' 


Since  v  and  d  are  unknown  C  is  unknown,  and  a  value  for  C  must  be  pro- 
visionally assumed. 

Assume  C  is  100  for  a  new  pipe  and  80  for  an  old  pipe,  and  solve  equation  (3) 
ford. 

From  (1)  find  v,  and  from  the  tables  find  the  value  of  C  corresponding  to  the 
values  of  d  and  v  thus  determined. 

If  C  differs  much  from  the  assumed  value,  recalculate  d  and  v  using  this  second 
value  of  C,  and  from  the  tables  find  a  third  value  for  C.  This  will  generally  be 
found  to  be  sufficiently  near  to  the  second  value  to  make  it  unnecessary  to  calculate 
d  and  v  a  third  time. 

The  approximation,  assuming  the  values  of  C  in  the  tables  are  correct,  can  be 
taken  to  any  degree  of  accuracy,  but  as  the  values  of  C  are  uncertain  it  will  not  as 
a  rule  be  necessary  to  calculate  more  than  two  values  of  d. 

yvnl 
Logarithmic  formula.     If  the  formula  li=!——  be  used,  d  can  be  found  direct, 

from 

p  log  d  =  nlog  v  +  logy  +  log  l-log  h. 

Example  3.  Find  the  diameter  of  a  steel  riveted  pipe,  which  will  discharge 
14  cubic  feet  per  second,  the  loss  of  head  by  friction  being  2  feet  per  mile.  It  is 
assumed  that  the  pipe  has  become  dirty  and  that  provisionally  C  =  110. 

From  equation  (3) 


2-55  . 14        /5280 

=-110— V2' 


or  f  log  d  =  log  16  -63, 

therefore  d  =  3-08  feet. 

For  a  thirty-eight  inch  pipe  Kuichling  found  C  to  be  113. 

The  assumption  that  C  is  110  is  nearly  correct  and  the  diameter  may  be  taken 
as  37  inches. 

Using  the  logarithmic  formula 

•00045V1-95; 
h= 


FLOW  THROUGH   PIPES 


153 


and  substituting  for  v  the  value  -—2- 


/,r\ 

) 


from  which 

5  -15  log  d = log  -00045  - 1  -95  log  0-7854  + 1-95  log  14  +  log  2640, 
and  d  =  3-07  feet. 

Short  pipe.     If  the  pipe  is  short  so  that  the  velocity  head  and  the  head  lost  at 
entrance  are  not  negligible  compared  with  the  loss  due  to  friction,  the  equation 

'    -0406Q2d      6-5/Q2 

"T"     :~c^T' 

when  a  value  is  given  to  C,  can  be  solved  graphically  by  plotting  two  curves 


Vi- 


•0406Q2 


.d  + 


6-5ZQ2 

~mr 


The  point  of  intersection  of  the  two  curves  will  give  the 
diameter  d. 

It  is  however  easier  to  solve  by  approximation  in  the 
following  manner. 

Neglect  the  term  in  d  and  solve  as  for  a  long  pipe. 

Choose  a  new  value  for  C  corresponding  to  this  ap- 
proximate diameter,  and  the  velocity  corresponding  to  it, 
and  then  plot  three  points  on  the  curve  y  =  d5,  choosing 
values  of  d  which  are  nearly  equal  to  the  calculated  value 
of  d,  and  two  points  of  the  straight  line 
_-Q406Q2d      6-5/Q2 
2/1  ~   ~~h~      "~C*}T'  T 

The  curve  y  =  d5  between  the  three  points  can  easily    •*  '5  •€ 

be  drawn,  as  in  Fig.  100,  and  where  the  straight  line  cuts  -pig.  100. 

the  curve,  gives  the  required  diameter.' 

Example  4.  One  hundred  and  twenty  cubic  feet  of  water  are  to  be  taken 
per  minute  from  a  tank  through  a  cast-iron  pipe  100  feet  long,  having  a  square- 
edged  entrance.  The  total  head  is  10  feet.  Find  the  diameter  of  the  pipe. 

Neglecting  the  term  in  d  and  assuming  C  to  be  100, 


and 


Therefore 


100  . 100 . 10 
d=  -4819  feet. 
2 


|(-4819)2 


=  10-9  ft.  per  sec. 


From  Table  XII,  the  value  of  C  is  seen  to  be  about  106  for  these  values  of 
d  and  v. 

A  second  value  for  d5  is 

6-5. 100. 4 
"    1062.10   " 
from  which  d=-476'. 

The  schedule  shows  the  values  of  d5  and  y  for  values  of  d  not  very  different 
from  the  calculated  value,  and  taking  C  as  106. 

d  -4  -5  -6 

d5  -01024  -03125  -0776 

y1  -0297  -0329 

The  line  and  curve  plotted  in  Fig.  100,  from  this  schedule,  intersect  at  4?  for  which 

d  =  -498  feet. 


154  HYDRAULICS 

It  is  seen  therefore  that  taking  106  as  the  value  of  C,  neglecting  the  term  in  d, 
makes  an  error  of  -022'  or  -264". 

This  problem  shows  that  when  the  ratio  —  is  about  200,  and  the  virtual  slope  is 
even  as  great  as  j1^,  for  all  practical  purposes,  the  friction  head  only  need  be  con- 
sidered. For  smaller  values  of  the  ratio  —  the  quantity  -025^  may  become  im- 

portant, but  to  what  extent  will  depend  upon  the  slope  of  the  hydraulic  gradient. 

The  logarithmic  formula  may  be  used  for  short  pipes  but  it  is  a  little  more 
cumbersome. 

Using  the  logarithmic  formula  to  express  the  loss  of  head  for  short  pipes  with 
square-edged  entrance, 


or  dan+i-26  _  .0406Q-d2n-2'75  =  . 

w 

When  suitable  values  are  given  to  7  and  ?i,  this  can  be  solved  by  plotting  the 
two  curves 

and  yl  =  - 


the  intersection  of  the  two  curves  giving  the  required  value  of  d. 

Problem  3.  To  find  wbat  the  discharge  between  the  reservoirs  of  problem  (1) 
would  be,  if  for  a  given  distance  Zj  the  pipe  , 

of  diameter  d  is  divided  into  two  branches  i  I 

laid  side  by  side  having  diameters  dl  and  d2,         [<  —    £   ---  >V  ---    £,     ---  ^ 
Fig-  101.  i  ! 

Assume  all  the  head  is  lost  in  friction.  A[  .       ty^j       d*          1^ 

Let  Qj  be  the  discharge  in  cubic  feet.  -CJ|  -  ^  --  K  f        *~ 

Then,  since  both  the  branches  BC  and  BD         |  -  1  --  ^   V 
are  connected  at  B  and  to  the  same  reservoir,  v^       ok          |  L) 

the  head  lost  in  friction  must  be  the  same  in 

BC  as  in  BD,  and  if  there  were  any  number  -    L   - 

of  branches  connected  at  B  the  head  lost  in  p-      -^Q-^ 

them  all  would  be  the  same. 

The  case  is  analogous  to  that  of  a  conductor  joining  two  points  between  which 
a  definite  difference  of  potential  is  maintained,  the  conductor  being  divided  between 
the  points  into  several  circuits  in  parallel. 

The  total  head  lost  between  the  reservoirs  is,  therefore,  the  head  lost  in  AB 
together  with  the  head  lost  in  any  one  of  the  branches. 

Let  v  be  the  velocity  in  AB,  vl  in  BC  and  vz  in  BD. 

Then  vd'2  =  vldl-  +  v2d^  ....................................  (1), 

and  the  difference  of  level  between  the  reservoirs 


And  since  the  head  lost  in  BC  is  the  same  as  in  BD,  therefore, 

4W-4W 

qtt  •-<«••• 

If  provisionally  Cj  be  taken  as  equal  to  C2  , 


FLOW   THROUGH   PIPES 


155 


Therefore, 


and 


v.d? 


___ 

A 
Vd, 


From  (2),  v  can  be  found  by  substituting  for  vl  from  (4),  and  thus  Q   can 
be  determined. 

If  AB,  BC,  and  CD  are  of  the  same  diameter  and  Zx  is  equal  to  Z2,  then 

and  h= 


or  Qi 

Problem  4.  Pipes  connecting  three  reservoirs.  As  in  Fig.  102,  let  three  pipes 
AB,  BC,  and  BD,  connect  three  reservoirs  A,  C,  D,  the  level  of  the  water  in  each 
of  which  remains  constant. 

Let  vl ,  v2,  and  v3  be  the  velocities  in  AB,  BC,  .and  BD  respectively,  Q15  Q2, 
and  Q3  the  quantities  flowing  along  these  pipes  in  cubic  feet  per  sec.,  llt  Z2,  and  13 
the  lengths  of  the  pipes,  and  dl ,  d2  and  d3  their  diameters. 


Fig.  102. 

Let  zlt  z2,  and  z3  be  the  heights  of  the  surfaces  of  the  water  in  the  reservoirs, 
and  z0  the  height  of  the  junction  B  above  some  datum. 
Let  hQ  be  the  pressure  head  at  B. 

Assume  all  losses,  other  than  those  due  to  friction  in  the  pipes,  to  be  negligible. 
The  head  lost  due  to  friction  for  the  pipe  AB  is 

A  1    _       O 

(1), 


•(2), 


andfor  the  pipe  BC, 

the  upper  or  lower  signs  being  taken,  according  as  to  whether  the  flow  is  from,  or 
towards,  the  reservoir  C. 

For  the  pipe  BD  the  head  lost  is 


z0  +  h0-zs (3). 

^3"«3 

Since  the  flow  from  A  and  C  must  equal  the  flow  into  D,  or  else  the  flow 
from  A  must  equal  the  quantity  entering  C  and  D,  therefore, 


or 


. 

There  are  four  equations,  from  which  four  unknowns  may  be  found,  if  it  is 
further  known  which  sign  to  take  in  equations  (2)  and  (4).  There  are  two  cases  to 
consider. 


156  HYDRAULICS 

Case  (a).  Given  the  levels  of  the  surfaces  of  the  water  in  the  reservoirs  and 
of  the  junction  B,  and  the  lengths  and  diameters  of  the  pipes,  to  find  the  quantity 
flowing  along  each  of  the  pipes. 

To  solve  this  problem,  it  is  first  necessary  to  obtain  by  trial,  whether  water  flows 
to,  or  from,  the  reservoir  C. 

First  assume  there  is  no  flow  along  the  pipe  BC,  that  is,  the  pressure  head  hQ  at 
B  is  equal  to  z2-z0. 

Then  from  (1),  substituting  for  vt  its  value          •  , 


_,      z 

..  !         2' 

= 


from  which  an  approximate  value  for  Q1  can  be  found.     By  solving  (3)  in  the  same 
way,  an  approximate  value  for  Q3 ,  is, 

CTT 


k 

If  Q3  is  found  to  be  equal  to  Q1?  the  problem  is  solved ;  but  if  Q3  is  greater  than 
Q!  ,  the  assumed  value  for  7z0  is  too  large,  and  if  less,  h0  is  too  small,  for  a  diminu- 
tion in  the  pressure  head  at  B  will  clearly  diminish  Q3  and  increase  QI}  and  will 
also  cause  flow  to  take  place  from  the  reservoir  C  along  CB.  Increasing  the 
pressure  head  at  B  will  decrease  Qlf  increase  Q3,  and  cause  flow  from  B  to  C. 

This  preliminary  trial  will  settle  the  question  of  sign  in  equations  (2)  and  (4) 
and  the  four  equations  may  be  solved  for  the  four  unknowns,  vl,  v2,  v3  and  h0.  It 
is  better,  however,  to  proceed  by  "trial  and  error." 

The  first  trial  shows  whether  it  is  necessary  to  increase  or  diminish  7?0  and  new 
values  are,  therefore,  given  to  h0  until  the  calculated  values  of  v1,  v2  and  v3  satisfy 
equation  (4). 

Case  (b).  Given  Q15  Q2,  Q3,  and  the  levels  of  the  surfaces  of  the  water  in 
the  reservoirs  and  of  the  junction  B,  to  find  the  diameters  of  the  pipes. 

In  this  case,  equation  (4)  must  be  satisfied  by  the  given  data,  and,  therefore, 
only  three  equations  are  given  from  which  to  calculate  the  four  unknowns  dlt 
d2,  ds  and  /z0.  For  a  definite  solution  a  fourth  equation  must  consequently  be 
found,  from  some  other  condition.  The  further  condition  that  may  be  taken  is 
that  the  cost  of  the  pipe  lines  shall  be  a  minimum. 

The  cost  of  pipes  is  very  nearly  proportional  to  the  product  of  the  length  and 
diameter,  and  if,  therefore,  1^  +  1^  + 1 sds  is  made  a  minimum,  the  cost  of  the 
pipes  will  be  as  small  as  possible. 

Differentiating,  with  respect  to  7z0 ,  the  condition  for  a  minimum  is,  that 

dd^     7  dd%     .  dd%     n  . 

l-i  -TT—  +  to  -r: \~  I")  ~rp  =0 ( 7 ). 

dh0        dh0      •  dh0 
Substituting  in  (1),  (2)  and  (3)  the  values  for  vlt  v2  and  v3, 


differentiating  and  substituting  in  (7) 


FLOW   THROUGH   PIPES  157 

Putting  the  values  of  Qlf  Q2,  and  Q3  in  (1),  (2),  (3),  and  (8),  there  are  four 
equations  as  before  for  four  unknown  quantities. 

It  will  be  better  however  to  solve  by  approximation. 

Give  some  arbitrary  value  to  say  d2,  and  calculate  h0  from  equation  (2). 

Then  calculate  dl  and  d3  by  putting  7i0  in  (1)  and  (3),  and  substitute  in 
equation  (8). 

If  this  equation  is  satisfied  the  problem  is  solved,  but  if  not,  assume  a  second 
value  for  d2  and  try  again,  and  so  on  until  such  values  of  dlt  d2,  d3  are  obtained 
that  (8)  is  satisfied. 

In  this,  as  in  simpler  systems,  the  pressure  at  any  point  in  the  pipes  ought  not 
to  fall  below  the  atmospheric  pressure. 

Flow  through  a  pipe  of  constant  diameter  when  the  flow  is  diminishing  at  a 
uniform  rate.  Let  I  be  the  length  of  the  pipe  and  d  its  diameter. 

Let  h  be  the  total  loss  of  head  in  the  pipe,  the  whole  loss  being  assumed  to  be 
by  friction. 

Let  Q  be  the  number  of  cubic  feet  per  second  that  enters  the  pipe  at  a  section  A, 
and  Q,  the  number  of  cubic  feet  that  passes  the  section  B,  I  feet  from  A,  the 
quantity  Q  -  Qx  being  taken  from  the  pipe,  by  branches,  at  a  uniform  rate  of 

~    l  cubic  feet  per  foot. 

Then,  if  the  pipe  is  assumed  to  be  continued  on,  it  is  seen  from  Fig.  103,  that 
if  the  rate  of  discharge  per  foot  length  of  the 
pipe  is  kept  constant,  the  whole  of  Q  will  be 
discharged  in  a  length  of  pipe, 

ZQ 

'"(Q-Qi)' 

The  discharge  past  any  section,  x  feet  from 
C,  will  be 

L  Fig.  103. 

The  velocity  at  the  section  is 


Assuming  that  in  an  element  of  length  dx  the  loss  of  head  due  to  friction  is 

7^/9* 

dm 
and  substituting  for  vx  its  value 

Qx 


the  loss  of  head  due  to  friction  in  the  length  I  is 


d1"25 
7     /  4Q 


If  Qx  is  zero,  I  is  equal  to  L,  and 

7     (4 

~?l  +  l    \7I 

The  result  is  simplified  by  taking  for  dh  the  value 


and  assuming  C  constant. 


158  HYDRAULICS 

Problem  5.  Pumping  water  through  long  pipes.  Required  the  diameter  of  a 
long  pipe  to  deliver  a  given  quantity  of  water,  against  a  given  effective  head,  in 
order  that  the  charges  on  capital  outlay  and  working  expenses  shall  be  a  minimum. 

Let  I  be  the  length  of  the  pipe,  d  its  diameter,  and  h  feet  the  head  against  which 
Q  cubic  feet  of  water  per  second  is  to  be  pumped. 

Let  the  cost  per  horse-power  of  the  pumping  plant  and  its  accommodation 
be  £N,  and  the  cost  of  a  pipe  of  unit  diameter  £n  per  foot  length. 

Let  the  cost  of  generating  power  be  £m  per  cent,  of  the  capital  outlay  in  the 
pumping  station,  and  the  interest,  depreciation,  and  cost  of  upkeep  of  the  pumping 
plant,  taken  together,  be  r  per  cent,  of  the  capital  outlay,  and  that  of  the  pipe  line 
r±  per  cent.  ;  r:  will  be  less  than  r.  The  horse-power  required  to  lift  the  water 
against  a  head  h  and  to  overcome  the  frictional  resistance  of  the  pipe  is 


Let  e  be  the  ratio  of  the  average  effective  horse-power  to  the  total  horse-power, 
including  the  stand-by  plant.     The  total  horse-power  of  the  plant  is  then 

„_>    0-1136Q  /,     6-48Q2r 
I      ~^~     r<2 

The  cost  of  the  pumping  plant  is  N  times  this  quantity. 
The  total  cost  per  year,  P,  of  the  station,  is 


Assuming  that  the  cost  of  the  pipe  line  is  proportional  to  the  diameter  and  to 
the  length,  the  capital  outlay  for  the  pipe  is,  £nld,  and  the  cost  of  upkeep  and 

.    £r,nld 
interest  is       '       . 

nrJd 


™        .  B  ., 

Therefore  0-1136  .—.-k+  --     + 


is  to  be  a  minimum. 

Differentiating  with  respect  to  d  and  equating  to  zero, 


and  d6=3*68(m+r)N.Q3 


That  is,  d  is  independent  of  the  length  I  and  the  head  against  which  the  water 
is  pumped. 


Taking  C  as  80,  e  as  0-6  and  as  50,  then 


•68  x  50       /Q 


80  x  80  x  -6 

=  0-603  */QT 
If(m±r)N 


and  if  is  25, 

w, 

d  =-535^/0^ 

Problem  6.  Pipe  with  a  nozzle  at  the  end.  Suppose  a  pipe  of  length  I  and 
diameter  D  has  at  one  end  a  nozzle  of  diameter  d,  through  which  water  is  dis- 
charged from  a  reservoir,  the  level  of  the  water  in  which  is  h  feet  above  the  centre 
of  the  nozzle. 

Required  the  diameter  of  the  nozzle  so  that  the  kinetic  energy  of  the  jet  is 
a  maximum. 


FLOW  THROUGH  PIPES  159 

Let  V  be  the  velocity  of  the  water  in  the  pipe. 

Then,  since  there  is  continuity  of  flow,  v  the  velocity  with  which  the  water 

leaves  the  nozzle  is  — ^—  . 
d2 

The  head  lost  by  friction  in  the  pipe  is 

4/ V-l  _  ±fvn .  d4 


The  kinetic  energy  of  the  jet  per  Ib.  of  flow  as  it  leaves  the  nozzle  is  — 

Therefore  *  =  |  (1  +  <£ )  

from  which  by  transposing  and  taking  the  square  root, 


The  weight  of  water  which  flows  per  second  —  —  d2.  v  .  w  where  M?  =  the  weight  of 

a  cubic  foot  of  water. 

Therefore,  the  kinetic  energy  of  the  jet,  is 


This  is  a  maximum  when  -^  =  0. 
ad 

Therefore 


.(3). 


4 

from  which  D5  +  4/W4  =  12/W4, 

and 

or 

v        "•/  " 

If  the  nozzle  is  not  circular  but  has  an  area  a,  then  since  in  the  circular  nozzle 
of  the  same  area 


16a2 
from  which  d4= 

Therefore  D 

and  a  =  0-278  /y- 

By  substituting  the  value  of  D5  from  (5)  in  (1)  it  is  at  once  seen  that,  »for 
maximum  kinetic  energy,  the  head  lost  in  friction  is 

iv2 

V  or  **• 

Problem  1.     Taking  the  same  data  as  in  problem  6,  to  find  the  area  of  the 
nozzle  that  the  momentum  of  the  issuing  jet  is  a  maximum. 

The  momentum  of  the  quantity  of  water  Q  which  flows  per  second,  as  it  leaves 

the  nozzle,  is  W  '  ^V  Ibs.  feet.     The  momentum  M  is,  therefore, 


9 

Substituting  for  ^  from  equation  (1),  problem  6, 


M= 


160  HYDRAULICS 

Differentiating,  and  equating  to  zero, 


and 


d=  \f  ^' 
64 


If  the  nozzle  has  an  area  a,     D5  = 


and  a=-3 

Substituting  for  D5  in  equation  (1)  it  is  seen  that  when  the  momentum  is  a 
maximum  half  the  head  h  is  lost  in  friction. 

Problem  6  has  an  important  application,  in  determining  the  ratio  of  the  size 
of  the  supply  pipe  to  the  orifice  supplying  water  to  a  Pelton  Wheel,  while  problem  7 
gives  the  ratio,  in  order  that  the  pressure  exerted  by  the  jet  on  a  fixed  plane 
perpendicular  to  the  jet  should  be  a  maximum. 

Problem  8.  Loss  of  head  due  to  friction  in  a  pipe,  the  diameter  of  which  varies 
uniformly.  Let  the  pipe  be  of  length  I  and  its  diameter  vary  uniformly  from  d0 
to  dj_. 

Suppose  the  sides  of  the  pipe  produced  until  they  meet  in  P,  Fig.  104. 

Then  ^=4  and  S  =  ^-    .  ...(1). 


The  diameter  of  the  pipe  at  any  distance  x  from  the  small  end  is 


The  loss  of  head  in  a  small  element  of  length  dx  is      2     ,  v  being  the  velocity 
when  the  diameter  is  d. 
T 


Fig.  104. 
If  Q  is  the  flow  in  cubic  ft.  per  second 


i* 

The  total  loss  of  head  h  in  a  length  I  is 

dx 


4  Q 

TTd2 


16Q2.  S5 


S5  /_!_  __  1      \ 

f  \s4    (s  +  ov 


Substituting  the  value  of  S  from  equation  (1)  the  loss  of  head  due  to  friction 
can  be  determined. 

Problem  9.  Pipe  line  consisting  of  a  number  of  pipes  of  different  diameters.  In 
practice  only  short  conical  pipes  are  used,  as  for  instance  in  the  limbs  of  a  Venturi 
meter. 

If  it  is  desirable  to  diminish  the  diameter  of  a  long  pipe  line,  instead  of  using 
a  pipe  the  diameter  of  which  varies  uniformly  with  the  length,  the  line  is  made  up 
of  a  number  of  parallel  pipes  of  different  diameters  and  lengths. 


FLOW  THROUGH  PIPES  161 

Let  Zj,  Z2,  13...  be  the  lengths  and  d^,  d2,  ds...  the  diameters  respectively,  of 
the  sections  of  the  pipe. 

The  total  loss  of  head  due  to  friction,  if  C  be  assumed  constant,  is 


The  diameter  d  of  the  pipe,  which,  for  the  same  total  length,  would  give  the 
same  discharge  for  the  same  loss  of  head  due  to  friction,  can  be  found  from  the 
equation 

-_  h    .    **       **    . 


The  length  L  of  a  pipe,  of  constant  diameter  D,  which  will  give  the  same 
discharge  for  the  same  loss  of  head  by  friction,  is 


Problem  10.  Pipe  acting  as  a  siphon.  It  is  sometimes  necessary  to  take  a 
pipe  line  over  some  obstruction,  such  as  a  hill,  which  necessitates  the  pipe  rising, 
not  only  above  the  hydraulic  gradient  as  in  Fig.  87,  but  even  above  the  original 
level  of  the  water  in  the  reservoir  from  which  the  supply  is  derived. 

Let  it  be  supposed,  as  in  Fig.  105,  that  water  is  to  be  delivered  from  the  reservoir 
B  to  the  reservoir  C  through  the  pipe  BAG,  which  at  the  point  A  rises  ^  feet  above 
the  level  of  the  surface  of  the  water  in  the  upper  reservoir. 


f 


Fig.  105. 

Let  the  difference  in  level  of  the  surfaces  of  the  water  in  the  reservoirs 
be  fe2  feet. 

Let  ha  be  the  pressure  head  equivalent  to  the  atmospheric  pressure. 

To  start  the  flow  in  the  pipe,  it  will  be  necessary  to  fill  it  by  a  pump  or  other 
artificial  means. 

Let  it  be  assumed  that  the  flow  is  allowed  to  take  place  and  is  regulated  so  that 
it  is  continuous,  and  the  velocity  v  is  as  large  as  possible. 

Then  neglecting  the  velocity  head  and  resistances  other  than  that  due  to  friction, 


/ 


L  and  d  being  the  length  and  diameter  of  the  pipe  respectively. 

The  hydraulic  gradient  is  practically  the  straight  line  DE. 

Theoretically  if  AF  is  made  greater  than  ha,  which  is  about  34  feet,  the  pressure 
at  A  becomes  negative  and  the  flow  will  cease. 

Practically  AF  cannot  be  made  much  greater  than  25  feet. 

To  find  the  maximum  velocity  possible  in  the  rising  limb  AB,  so  that  the  pressure 
head  at  A  shall  just  be  zero. 

Let  vm  be  this  velocity.     Let  the  datum  level  be  the  surface  of  the  water  in  C. 


L.  H. 


11 


162  HYDRAULICS 

But 
Therefore 


If  the  pressure  head  is  not  to  be  less  than  10  feet  of  water, 


limiting  velocity,  and  it  will  be  necessary  to  throttle  the  pipe  at  C  by  means  of  a 
valve,  so  as  to  keep  the  limb  AC  full  and  to  keep  the  "  siphon  "  from  being  broken. 

In  designing  such  a  siphon  it  is,  therefore,  necessary  to  determine  whether  the 
flow  through  the  pipe  as  a  whole  under  a  head  h2  is  greater,  or  less  than,  the  flow 
in  the  rising  limb  under  a  head  ha-  hl. 

If  AB  is  short,  or 
friction  in  AB  will  be 


2gd 

If  the  end  C  of  the  pipe  is  open  to  the  atmosphere  instead  of  being  connected  to 
a  reservoir,  the  total  head  available  will  be  h3  instead  of  h^. 

111.  Velocity  of  flow  in  pipes. 

The  mean  velocity  of  flow  in  pipes  is  generally  about  3  feet 
per  second,  but  in  pipes  supplying  water  to  hydraulic  machines, 
and  in  short  pipes,  it  may  be  as  high  as  10  feet  per  second. 

If  the  velocity  is  high,  the  loss  of  head  due  to  friction  in  long 
pipes  becomes  excessive,  and  the  risk  of  broken  pipes  and  valves 
through  attempts  to  rapidly  check  the  flow,  by  the  sudden  closing 
of  valves,  or  other  causes,  is  considerably  increased. 

On  the  other  hand,  if  the  velocity  is  too  small,  unless  the  water 
is  very  free  from  suspended  matter,  sediment*  tends  to  collect  at 
the  lower  parts  of  the  pipe,  and  further,  at  low  velocities  it  is 
probable  that  fresh  water  sponges  and  polyzoa  will  make  their 
abode  on  the  surface  of  the  pipe,  and  thus  diminish  its  carrying 
capacity. 

112.  Transmission   of  power    along   pipes   by  hydraulic 
pressure. 

Power  can  be  transmitted  hydraulically  through  a  considerable 
distance,  with  very  great  efficiency,  as  at  high  pressures  the  per 
centage  loss  due  to  friction  is  small. 

Let  water  be  delivered  into  a  pipe  of  diameter  d  feet  under  a 
head  of  H  feet,  or  pressure  of  p  Ibs.  per  sq.  foot,  for  which  the 

equivalent  head  is  H  =  —  feet. 
w 

*  An  interesting  example  of  this  is  quoted  on  p.  82  Trans.  Am.S.G.E. 
Vol.  XLIV. 


FLOW  THROUGH   PIPES  163 

Let  the  velocity  of  flow  be  v  feet  per  second,  and  the  length  of 
the  pipe  L  feet. 

The  head  lost  due  to  friction  is 


-f    ...........................  «>, 

and  the  energy  per  pound  available  at  the  end  of  the  pipe  is, 
therefore, 

' 


vl.  C\           t         • 

w  2gd 
The  efficiency  is 

H-fr  h 

H  H 


The  fraction  of  the  given  energy  lost  is 

h 

m  =  H' 

For  a  given  pipe  the  efficiency  increases  as  the  velocity 
diminishes. 

If  /  and  L  are  supposed  to  remain  constant,  the  efficiency  is 

i?2 
constant  if   -^FT  is  constant,  and  since  v  is  generally  fixed  from 

other  conditions  it  may  be  supposed  constant,  and  the  efficiency 
then  increases  as  the  product  dH.  increases. 

If  W  is  the  weight  of  water  per  second  passing  through  the 
pipe,  the  work  put  into  the  pipe  is  W  .  H  foot  Ibs.  per  second,  the 
available  work  per  second  at  the  end  of  the  pipe  is  W  (H  -  h),  and 
the  horse-power  transmitted  is 

Hp_W.(H-/Q     WH 

~     =1~ 

Since 


„ 
the  horse-power         =     --    H- 


,-,        /,  dmS. 

therefore,  v  =  4*01 


and  the  horse-power   =  0'357  *         -  d?~H.%  (1  -  m). 


11—2 


164  HYDRAULICS 

If  p  is  the  pressure  per  sq.  inch 


and  the  horse-power     =  1'24  */  -^  d*p^  (1  -  m)  . 


From  this  equation  if  m  is  given  and  L  is  known  the  diameter  d 
to  transmit  a  given  horse-power  can  be  found,  and  if  d  is  known  the 
longest  length  L  that  the  loss  shall  not  be  greater  than  the  given 
fraction  m  can  be  found. 

The  cost  of  the  pipe  line  before  laying  is  proportional  to  its 
weight,  and  the  cost  of  laying  approximately  proportional  to  its 
diameter. 

If  t  is  the  thickness  of  the  pipe  in  inches  the  weight  per  foot 
length  is  8l'5irdt  Ibs.,  approximately. 

Assuming  the  thickness  of  the  pipe  to  be  proportional  to  the 
pressure,  i.e.  to  the  head  H, 

t  =  kp  =  &H, 

and  the  weight  per  foot  may  therefore  be  written 

w  =  Jcid  .  H. 

The  initial  cost  of  the  pipe  per  foot  will  then  be 

C=&2MH  =  K.d.H, 

and  since  the  cost  of  laying  is  approximately  proportional  to  d, 
the  total  cost  per  foot  is 

P-K.d.H  +  K^. 
And  since  the  horse-power  transmitted  is 

HP  =  '357  d*H*  (1  -  m), 


for  a  given  horse-power  and  efficiency,  the  initial  cost  per  horse- 
power including  laying  will  be  a  minimum  when 

0^7./i-d»H*(l-«) 


is  a  maximum. 

In  large  works,  docks,  and  goods  yards,  the  hydraulic  trans- 
mission of  power  to  cranes,  capstans,  riveters  and  other  machines 
is  largely  used. 

A  common  pressure  at  which  water  is  supplied  from  the  pumps 
is  700  to  750  Ibs.  per  sq.  inch,  but  for  special  purposes,  it  is 
sometimes  as  high  as  3000  Ibs.  per  sq.  inch.  These  high  pressures 
are,  however,  frequently  obtained  by  using  an  intensifier  (Ch.  XI) 
to  raise  the  ordinary  pressure  of  700  Ibs.  to  the  pressure  required. 


FLOW   THROUGH   PIPES  165 

The  demand  for  hydraulic  power  for  the  working  of  lifts,  etc. 
has  led  to  the  laying  down  of  a  network  of  mains  in  several  of  the 
large  cities  of  Great  Britain.  In  London  a  mean  velocity  of  4  feet 
per  second  is  allowed  in  the  mains  and  the  pressure  is  750  Ibs. 
per  sq.  inch.  In  later  installations,  pressures  of  1100  Ibs.  per 
sq.  inch  are  used. 

113.    The  limiting  diameter  of  cast-iron  pipes. 

The  diameter  d  for  a  cast-iron  pipe  cannot  be  made  very  large 
if  the  pressure  is  high. 

If  p  is  the  safe  internal  pressure  per  sq.  inch,  and  s  the  safe 
stress  per  sq.  inch  of  the  metal,  and  rx  and  r2  the  internal  and 
external  radii  of  the  pipe, 


- 

rf+rf 

For  a  pressure  p  =  1000  Ibs.  per  sq.  inch,  and  a  stress  s  of 
3000  Ibs.  per  sq.  inch,  r2  is  5*65  inches  when  r\  is  4  inches,  or  the 
pipe  requires  to  be  1'65  inches  thick. 

If,  therefore,  the  internal  diameter  is  greater  than  8  inches,  the 
pipe  becomes  very  thick  indeed. 

The  largest  cast-iron  pipe  used  for  this  pressure  is  between 
7"  and  8"  internal  diameter. 

Using  a  maximum  velocity  of  5  feet  per  second,  and  a  pipe 
7  J  inches  diameter,  the  maximum  horse-power,  neglecting  friction, 
that  can  be  transmitted  at  1000  Ibs.  per  sq.  inch  by  one  pipe  is 
T^_  4418x1000x5 

550 
=  400. 

The  following  example  shows  that,  if  the  pipe  is  13,300  feet 
long,  15  per  cent,  of  the  power  is  lost  and  the  maximum  power 
that  can  be  transmitted  with  this  length  of  pipe  is,  therefore, 
320  horse-power. 

Steel  mains  are  much  more  suitable  for  high  pressures,  as  the 
working  stress  may  be  as  high  as  7  tons  per  sq.  inch.  The  greater 
plasticity  of  the  metal  enables  them  to  resist  shock  more  readily 
than  cast-iron  pipes  and  slightly  higher  velocities  can  be  used. 

A  pipe  15  inches  diameter  and  J  inch  thick  in  which  the 
pressure  is  1000  Ibs.  per  sq.  inch,  and  the  velocity  5  ft.  per  second, 
is  able  to  transmit  1600  horse-power. 

Example.  Power  is  transmitted  along  a  cast-iron  main  7£  inches  diameter  at 
a  pressure  of  1000  Ibs.  per  sq.  inch.  The  velocity  of  the  water  is  5  feet  per  second. 

Find  the  maximum  distance  the  power  can  be  transmitted  so  that  the  efficiency 
is  uot  less  than  85°/0. 

*  Ewing's  Strength  of  Materials. 


166 


therefore 

Then 
from  which 


HYDRAULICS 
d  =  0-625  feet, 


345'  = 


=  0-15x2300 
=  345  feet. 
4  x  0-0104  x  25  .  L 


L  = 


2.9  x  0-625 
345  x  64-4  x  0-625 


0-0104  x  100 
=  13,300  feet. 


D 


114.    Pressures  on  pipe  bends. 

If  a  bent  pipe  contain  a  fluid  at  rest,  the  intensity  of  pressure 
being  the  same  in  all  directions, 
the  resultant  force  tending  to  move 
the  pipe  in  any  direction  will  be 
the  pressure  per  unit  area  multiplied 
by  the  projected  area  of  the  pipe 
on  a  plane  perpendicular  to  that 
direction. 

If  one  end  of  a  right-angled 
elbow,  as  in  Fig.  106,  be  bolted  to 
a  pipe  full  of  water  at  a  pressure  p 


Fig.  106. 


Fig.  107. 


pounds  per  sq.  inch  by  gauge,  and  on  the  other  end  of  the  elbow 
is  bolted  a  flat  cover,  the  tension  in  the  bolts  at  A  will  be  the 
same  as  in  the  bolts  at  B.  The  pressure  on  the  cover  B  is  clearly 
'7854p^2,  d  being  the  diameter  of  the  pipe  in  inches.  If  the  elbow 
be  projected  on  to  a  vertical  plane  the  projection  of  ACB  is  daefc, 
the  projection  of  DEF  is  abcfe.  The  resultant  pressure  on  the 
elbow  in  the  direction  of  the  arrow  is,  therefore,  p  .  abed  =  *7854p<i2. 

If  the  cover  B  is  removed,  and  water  flows  through  the  pipe 
with  a  velocity  v  feet  per  second,  the  horizontal  momentum  of  the 
water  is  destroyed  and  there  is  an  additional  force  in  the  direction 
of  the  arrow  equal  to  '7854w^V. 

When  flow  is  taking  place  the  vertical  force  tending  to  lift  the 
elbow  or  to  shear  the  bolts  at  A  is 


If  the  elbow  is  less  than  a  right 
angle,    as  in  Fig.   108,  the    total 
tension  in  the  bolts  at  A  is 
T  =  p  (daehgc  -  aefgc) 


e 


A 

9? 


Fig.  108. 


and  since  the  area  aehgcb  is  common  to  the  two  projected  areas, 
T  =  '7854d2  (p  -  p  cos  0  +  wv*  cos  0)  . 


FLOW   THROUGH   PIPES 


167 


Consider  now  a  pipe  bent  as  shown  in  Fig.  109,  the  limbs  AA 
and  FF  being  parallel,  and  the  water  being  supposed  at  rest. 

The  total  force  acting  in  the  direction  A  A  is 

P  =  p {dcghea - aefgcb  +  d'cg'h'e'a  - a'ef'g'cb'}, 
which  clearly  is  equal  to  0. 


If  now  instead  of  the  fluid  being  at  rest  it  has  a  uniform 
velocity,  the  pressure  must  remain  constant,  and  since  there  is  no 
change  of  velocity  there  is  no  change  of  momentum,  and  the  re- 
sultant pressure  in  the  direction  parallel  to  AA  is  still  zero. 

There  is  however  a  couple  acting  upon  the  bend  tending  to 
rotate  it  in  a  clockwise  direction. 

Let  p  and  q  be  the  centres  of  gravity  of  the  two  areas  daehgc 
and  aefgcb  respectively,  and  m  and  n  the  centres  of  gravity  of 
d'a'e'lig'c   and  aefgcb'. 

Through  these  points  there  are  parallel  forces  acting  as  shown 
by  the  arrows,  and  the  couple  is 

M  =  P' .  mn  -  P .  pq. 

The  couple  P .  pq  is  also  equal  to  the  pressure  on  the  semicircle 
adc  multiplied  by  the  distance  between  the  centres  of  gravity  of 
adc  and  efg,  and  the  couple  P' .  mn  is  equal  to  the  pressure  on  a'd'c 
multiplied  by  the  distance  between  the  centres  of  gravity  of  a'd'c 
and  e'f'g'. 

Then  the  resultant  couple  is  the  pressure  on  the  semicircle  efg 
multiplied  by  the  distance  between  the  centres  of  gravity  of  efg 
and  e'f'g'. 

If  the  axes  of  FF  and  AA  are  on  the  same  straight  line  the 
couple,  as  well  as  the  force,  becomes  zero. 

It  can  also  be  shown,  by  similar  reasoning,  that,  as  long  as  the 
diameters  at  F  and  A  are  equal,  the  velocities  at  these  sections 
being  therefore  equal,  and  the  two  ends  A  and  F  are  in  the  same 
straight  line,  the  force  and  the  couple  are  both  zero,  whatever  the 
form  of  the  pipe.  If,  therefore,  as  stated  by  Mr  Froude,  "the 


168  HYDRAULICS 

two  ends  of  a  tortuous  pipe  are  in  the  same  straight  line,  there  is 
no  tendency  for  the  pipe  to  move." 

115.     Pressure  on  a  plate  in  a  pipe  filled  with  flowing  water. 

The  pressure  on  a  plate  in  a  pipe  filled  with  flowing  water,  with 
its  plane  perpendicular  to  the  direction  of  flow,  on  certain  assump- 
tions, can  be  determined. 

Let  PQ,  Fig.  110,  be  a  thin  plate  of  area  a  and  let  the  sectional 
area  of  the  pipe  be  A. 

The  stream  as  it  passes  the  edge  of         fL gj  & 

the  plate  will  be  contracted,  and  the 
section  of  the  stream  on  a  plane  gd  will 
be  c(A-a),  c  being  some  coefficient  of 
contraction. 

It  has  been  shown  on  page  52  that 
for  a  sharp-edged  orifice  the  coefficient 
of  contraction  is  about  0*625,  and  when 

part  of  the  orifice  is  fitted  with  sides  so  that  the  contraction  is 
incomplete  and  the  stream  lines  are  in  part  directed  perpendi- 
cular to  the  orifice,  the  coefficient  of  contraction  is  larger. 

If  a  coefficient  in  this  case  of  0'66  is  assumed,  it  will  probably 
be  not  far  from  the  truth. 

Let  YI  be  the  velocity  through  the  section  gd  and  V  the  mean 
velocity  in  the  pipe. 

The  loss  of  head  due  to  sudden  enlargement  from  gd  to  ef  is 

(Vi-V)2 

2<7 

Let  the  pressures  at  the  sections  ab,  gd,  ef  be  p,  pi  and  p2  pounds 
per  square  foot  respectively. 

Bernouilli's  equations  for  the  three  sections  are  then, 


w     2g     w      2g 

and  5*^5*5*^?- (2)- 

Adding  (1)  and  (2) 

(2  _  P*\  =  (Vi-y)2 
\w      w/  2g 

The  whole  pressure  on  the  plate  in  the  direction  of  motion  is  then 


Y2 
=  w  .a 


'2g  V66(A-a] 


FLOW  THROUGH   PIPES 


169 


If  a  =  JA, 


va 

~-  nearly. 


0-46.g.Y2 


116.     Pressure  on  a  cylinder. 

If  instead  of  a  thin  plate  a  cylinder  be  placed  in  the  pipe, 
with  its  axis  coincident  with  the  axis  of  the  pipe,  Fig.  Ill,  there 
are  two  enlargements  of  the  section  of  the  water. 

As  the  stream  passes  the  up-stream  edge  of  the  cylinder,  it 
contracts  to  the  section  at  cd,  and  then  enlarges  to  the  section 
ef.  It  again  enlarges  at  the  down-stream  end  of  the  cylinder 
from  the  section  ef  to  the  section  gh. 

cu 


g^ 


Fig.  111. 

Let  Vi,  v2j  v3)  v*  be  the  velocities  at  ab,  cd,  ef  and  gh  re- 
spectively, v4  and  t;i  being  equal. 

Between  cd  and  ef  there  is  a  loss  of  head 


and  between  ef  and  gh  there  is  a  loss  of 


2<7 

The  Bernouilli's  equations  for  the  sections  are 


Pi  +  v±  =  P?+'? 
w     Zg     w     2g' 


Adding  (2)  and  (3), 


+       =  .  = 

w     2g     w     2g     w     2g          2g 

P3^=P4+^L 
w     2g     w     2g 


2g 


(1), 
(2), 
(3). 


2g 


170  HYDRAULICS 

If  the  coefficient  of  contraction  at  cd  is  c,  the  area  at  cd 

A  —  a 


A 


Then 


c.(A-a)    '  ~A-a' 

Therefore 

A 


and  the  pressure  on  the  cylinder  is 

P  =  (Pi  -  pO  •  «. 


EXAMPLES. 

(1)  A  new  cast-iron  pipe  is  2000  ft.  long  and  6  ins.  diameter.     It  is  to 
discharge  50  c.  ft.  of  water  per  minute.    Find  the  loss  of  head  in  friction 
and  the  virtual  slope. 

(2)  What  is  the  head  lost  per  mile  in  a  pipe  2  ft.  diameter,  discharging 
6,000,000  gallons  in  24  hours  ?    /=  '007. 

(3)  A  pipe  is  to  supply  40,000  gallons  in  24  hours.     Head  of  water 
above  point  of  discharge  =  36  ft.     Length  of  pipe  =  2^  miles.     Find  its 
diameter.     Take  C  from  Table  XII. 

(4)  A  pipe  is  12  ins.  in  diameter  and  3  miles  in  length.     It  connects 
two  reservoirs  with  a  difference  of  level  of  20  ft.     Find  the  discharge  per 
minute  in  c.  ft.     Use  Darcy's  coefficient  for  corroded  pipes. 

(5)  A  water  main  has  a  virtual  slope  of  1  in  900  and  discharges  35  c.  ft. 
per  second.     Find  the  diameter  of  the  main.     Coefficient  /  is  0'007. 

(6)  A  pipe  12  ins.  diameter  is  suddenly  enlarged  to  18  ins.,  and  then  to 
24  ins.  diameter.     Each  section  of  pipe  is  100  feet  long.     Find  the  loss  of 
head  in  friction  in  each  length,  and  the  loss  due  to  shock  at  each  enlarge- 
ment.    The  discharge  is  10  c.  ft.  per  second,  and  the  coefficient  of  friction 
/=  '0106.     Draw,  to  scale,  the  hydraulic  gradient  of  the  pipe. 

(7)  Find  an  expression  for  the  relative  discharge  of  a  square,  and  a 
circular  pipe  of  the  same  section  and  slope. 

(8)  A  pipe  is  6  ins.  diameter,  and  is  laid  for  a  quarter  mile  at  a  slope 
of  1  in  50;  for  another  quarter  mile  at  a  slope  of  1  in  100;  and  for  a  third 
quarter  mile  is  level.     The  level  of  the  water  is  20  ft.  above  the  inlet  end, 
and  9  ft.  above  the  outlet  end.     Find  the  discharge  (neglecting  all  losses 
except  skin  friction)  and  draw  the  hydraulic  gradient.     Mark  the  pressure 
in  the  pipe  at  each  quarter  mile. 

(9)  A  pipe  2000  ft.  long  discharges  Q  c.  ft.  per  second.     Find  by  how 
much  the  discharge  would  be  increased  if  to  the  last  1000  ft.  a  second  pipe 
of  the  same  size  were  laid  alongside  the  first  and  the  water  allowed  to  flow 
equally  well  along  either  pipe. 


FLOW  THROUGH   PIPES  171 

(10)  A  reservoir,  the  level  of  which  is  50  ft.  above  datum,  discharges 
into  a  second  reservoir  30  ft.  above  datum,  through  a  12  in.  pipe,  5000  ft. 
in  length ;  find  the  discharge.     Also,  taking  the  levels  of  the  pipe  at  the 
upper  reservoir,  and  at  each  successive  1000  ft.,  to  be  40,  25,  12,  12,  10,  15, 
ft.  above  datum,  write  down  the  pressure  at  each  of  these  points,  and 
sketch  the  position  of  the  line  of  hydraulic  gradient. 

(11)  It  is  required  to  draw  off  the  water  of  a  reservoir  through  a 
pipe  placed  horizontally.     Diameter  of  pipe  6  ins.     Length  40  ft.     Ef- 
fective head  20  ft.     Find  the  discharge  per  second. 

(12)  Given  the  data  of  Ex.  11  find  the  discharge,  taking  into  account 
the  loss  of  head  if  the  pipe  is  not  bell-mouthed  at  either  end. 

(13)  A  pipe  4  ins.  diameter  and  100  ft.  long  discharges  \  c.  ft.  per 
second.  Find  the  head  expended  in  giving  velocity  of  entry,  in  overcoming 
mouthpiece  resistance,  and  in  friction. 

(14)  Required  the  diameter  of  a  pipe  having  a  fall  of  10  ft.  per  mile, 
and  capable  of  delivering  water  at  a  velocity  of  3  ft.  per  second  when  dirty. 

(15)  Taking  the  coefficient  /  as  O'Ol  (l  +  ?oj)»  nn^  h°w  much  water 

would  be  discharged  through  a  12 -inch  pipe  a  mile  long,  connecting  two 
reservoirs  with  a  difference  of  level  of  20  feet. 

(16)  Water  flows  through  a  12-inch  pipe  having  a  virtual  slope  of  3  feet 
per  1000  feet  at  a  velocity  of  3  feet  per  second. 

Find  the  friction  per  sq.  ft.  of  surface  of  pipe  in  Ibs. 

Also  the  value  of  /  in  the  ordinary  formula  for  flow  in  pipes. 

(17)  Find  the  relative  discharge  of  a  6-inch  main  with  a  slope  of 
1  in  400,  and  a  4 -inch  main  with  a  slope  of  1  in  50. 

(18)  A  6-inch  main  7  miles  in  length  with  a  virtual  slope  of  1  in  100 
is  replaced  by  4  miles  of  6-inch  main,  and  3  miles  of  4-inch  main.     Would 
the  discharge  be  altered,  and,  if  so,  by  how  much  ? 

(19)  Find  the  velocity  of  flow  in  a  water  main  10  miles  long,  con- 
necting two  reservoirs  with  a  difference  of  level  of  200  feet.     Diameter  of 
main  15  inches.     Coefficient /=0'009. 

(20)  Find  the  discharge,  if  the  pipe  of  the  last  question  is  replaced  for 
the  first  5  miles  by  a  pipe  20  inches  diameter  and  the  remainder  by  a  pipe 
12  inches  diameter. 

(21)  Calculate  the  loss  of  head  per  mile  in  a  10-inch  pipe  (area  of  cross 
section  0'54  sq.  ft.)  when  the  discharge  is  2£  c.  ft.  per  second. 

(22)  A  pipe  consists  of  £  a  mile  of  10  inch,  and  £  a  mile  of  5 -inch  pipe, 
and  conveys  2£  c.  ft.  per  second.     State  from  the  answer  to  the  previous 
question  the  loss  of  head  in  each  section  and  sketch  a  hydraulic  gradient. 
The  head  at  the  outlet  is  5  ft. 

(23)  What  is  the  head  lost  in  friction  in  a  pipe  3  feet  diameter 
discharging  6,000,000  gallons  in  12  hours? 

(24)  A  pipe  2000  feet  long  and  8  inches  diameter  is  to  discharge  85  c.  ft. 
per  minute.     What  must  be  the  head  of  water  ? 


172  HYDRAULICS 

(25)  A  pipe  6  inches  diameter,  50  feet  long,  is  connected  to  the  bottom 
of  a  tank  50  feet  long  by  40  feet  wide.  The  original  head  over  the  open 
end  of  the  pipe  is  15  feet.  Find  the  time  of  emptying  the  tank,  assuming 
the  entrance  to  the  pipe  is  sharp-edged. 

If  ft = the  head  over  the  exit  of  the  pipe  at  any  moment, 
h_v^      'JM      4/t?250' 
2gr       2g      2g  x  0*5' 

from  which,  v=l^mf 

In  time  dt,  the  discharge  is 


144 
In  time  dt  the  surface  falls  an  amount  dh. 

Therefore 

Integrating, 

2000  (1-5  +400/)0    /—     79000  (1-5  +400/) 

C  =  -  ;  -  -   A   V  lO  =     -  ;  -  - 

0-196  \/20  \/20 

(26)  The  internal  diameter  of  the  tubes  of  a  condenser  is  0'654  inches. 
The  tubes  are  7  feet  long  and  the  number  of  tubes  is  400.     The  number  of 
gallons  per  minute  flowing  through  the  condenser  is  400.     Find  the  loss  of 
head  due  to  friction  as  the  water  flows  through  the  tubes.    /=0'006. 

(27)  Assuming  fluid  friction  to  vary  as  the  square  of  the  velocity,  find 
an  expression  for  the  work  done  in  rotating  a  disc  of  diameter  d  at  an 
angular  velocity  a  in  water. 

(28)  What  horse-power  can  be  conveyed  through  a  6-in.  main  if  the 
working  pressure  of  the  water  supplied  from  the  hydraulic  power  station  is 
700  Ibs.  per  sq.  in.?     Assume  that  the  velocity  of  the  water  is  limited  to 
3  ft.  per  second. 

(29)  Ten  horse-power  is  to  be  transmitted  by  hydraulic  pressure  a 
distance  of  a  mile.    Find  the  diameter  of  pipe  and  pressure  required  for  an 
efficiency  of  '9  when  the  velocity  is  5  ft.  per  sec. 

The  frictional  loss  is  given  by  equation 

„  v*      4Z 
h  =  '0l7r  .   -T  . 
2g      d 

(30)  Find  the  inclination  necessary  to  produce  a  velocity  of  4^  feet  per 
second  in  a  steel  water  main  31  inches  diameter,  when  running  full  and 
discharging  with  free  outlet,  using  the  formula 

. 


(31)     The  following  values  of  the  slope  i  and  the  velocity  v  were 
determined  from  an  experiment  on  flow  in  a  pipe  '1296  ft.  diam. 
i    -00022     -00182       -00650       '02389       -04348     -12315       -22408 
v     -205         -606         1-252         2'585         3'593       6'310        8-521 


FLOW  THROUGH   PIPES  173 

Determine  k  and  n  in  the  formula 

i=kvn. 

Also  determine  values  of  C  for  this  pipe  for  velocities  of  -5,  1,  3,  5  and 
7  feet  per  sec. 

(32)  The  total  length  of  the  Coolgardie  steel  aqueduct  is  307£  miles 
and  the  diameter  30  inches.     The  discharge  per  day  may  be  5,600,000 
gallons.     The  water  is  lifted  a  total  height  of  1499  feet. 

Find     (a)     the  head  lost  due  to  friction, 

(6)     the  total  work  done  per  minute  in  raising  the  water. 

(33)  A  pipe  2  feet  diameter  and  500  feet  long  without  bends  furnishes 
water  to  a  turbine.     The  turbine  works  under  a  head  of  25  feet  and  uses 
10  c.  ft.  of  water  per  second.     What  percentage  of  work  of  the  fall  is  lost 

n  friction  in  the  pipe  ? 

Coefficient  /=  '007  (  1  +        \  . 


(34)  Eight  thousand  gallons  an  hour  have  to  be  discharged  through 
each  of  six  nozzles,  and  the  jet  has  to  reach  a  height  of  80  ft. 

If  the  water  supply  is  l£  miles  away,  at  what  elevation  above  the 
nozzles  would  you  place  the  required  reservoir,  and  what  would  you 
make  the  diameter  of  the  supply  main  ? 

Give  the  dimensions  of  the  reservoir  you  would  provide  to  keep  a 
constant  supply  for  six  hours.  Lond.  Un.  1903. 

(35)  The  pipes  laid  to  connect  the  Vyrnwy  dam  with  Liverpool  are 
42  inches  diameter.     How  much  water  will  such  a  pipe  supply  in  gallons 
per  diem  if  the  slope  of  the  pipe  is  4^  feet  per  mile  ? 

At  one  point  on  the  line  of  pipes  the  gradient  is  6|  feet  per  mile,  and  the 
pipe  diameter  is  reduced  to  39  inches  ;  is  this  a  reasonable  reduction  in  the 
dimension  of  the  cross  section  ?  Lond.  Un.  1905. 

(36)  Water  under  a  head  of  60  feet  is   discharged  through  a  pipe 
6  inches  diameter  and  150  feet  long,  and  then  through  a  nozzle  the  area  of 
which  is  one-tenth  the  area  of  the  pipe.   Neglecting  all  losses  except  friction, 
find  the  velocity  with  which  the  water  leaves  the  nozzle. 

(37)  Two  rectangular  tanks  each  50  feet  long  and  50  feet  broad  are 
connected  by  a  horizontal  pipe  4  inches  diameter,  1000  feet  long.     The 
head  over  the  centre  of  the  pipe  at  one  tank  is  12  feet,  and  over  the  other 
4  feet  when  flow  commences. 

Determine  the  time  taken  for  the  water  in  the  two  tanks  to  come  to  the 
same  level.  Assume  the  coefficient  C  to  be  constant  and  equal  to  90. 

(38)  Two  reservoirs  are  connected  by  a  pipe   1  mile  long  and  10" 
diameter;    the    difference    in    the    water    surface    levels    being    25    ft. 


Determine  the  flow  through  the  pipe  in  gallons  per  hour  and  find  by 
how  much  the  discharge  would  be  increased  if  for  the  last  2000  ft.  a  second 
pipe  of  10"  diameter  is  laid  alongside  the  first.  Lond.  Un.  1905. 

(39)  A  pipe  18"  diameter  leads  from  a  reservoir,  300  ft.  above  the 
datum,  and  is  continued  for  a  length  of  5000  ft.  at  the  datum,  the  length 
being  15,000  ft.  For  the  last  5000  ft.  of  its  length  water  is  drawn  off  by 


174  HYDRAULICS 

service  pipes  at  the  rate  of  10  c.  ft.  per  min.  per  500  ft.  uniformly.    Find 
the  pressure  at  the  end  of  the  pipe.    Lond.  Un.  1906. 

(40)  350  horse-power  is  to  be  transmitted  by  hydraulic  pressure  a 
distance  of  1^  miles. 

Find  the  number  of  6  ins.  diameter  pipes  and  the  pressure  required  for 
an  efficiency  of  92  per  cent.    /='01.     Take  v  as  3  ft.  per  sec. 

(41)  Find  the  loss  of  head  due  to  friction  in  a  water  main  L  feet  long, 
which  receives  Q  cubic  feet  per  second  at  the  inlet  end  and  delivers 

j-  cubic  feet  to  branch  mains  for  each  foot  of  its  length. 
What  is  the  form  of  the  hydraulic  gradient  ? 

(42)  A  reservoir  A  supplies  water  to  two  other  reservoirs  B  and  C. 
The  difference  of  level  between  the  surfaces  of  A  and  B  is  75  feet,  and 
between  A  and  C  97'5  feet.     A  common  8-inch  cast-iron  main  supplies  for 
the  first  850  feet  to  a  point  D.     A  6-inch  main  of  length  1400  feet  is  then 
carried  on  in  the  same  straight  line  to  B,  and  a  5  -inch  main  of  length 
630  feet  goes  to  C.     The  entrance  to  the  8-inch  main  is  bell-mouthed,  and 
losses  at  pipe  exits  to  the  reservoirs  and  at  the  junction  may  be  neglected. 
Find  the  quantity  discharged  per  minute  into  the  reservoirs  B  and  C. 
Take  the  coefficient  of  friction  (/)  as  '01.     Lond.  Un.  1907. 

(43)  Describe  a  method  of  finding  the  "loss  of  head"  in  a  pipe  due  to 
the  hydraulic  resistances,  and  state  how  you  would  proceed  to  find  the 
loss  as  a  function  of  the  velocity. 

(44)  A  pipe,  Z  feet  long  and  D  feet  in  diameter,  leads  water  from  a 
tank  to  a  nozzle  whose  diameter  is  d,  and  whose  centre  is  h  feet  below 
the  level  of    water  in  the  tank.     The  jet  impinges  on  a  fixed  plane 
surface.     Assuming  that  the  loss  of  head  due  to  hydraulic  resistance  is 
given  by 


show  that  the  pressure  on  the  surface  is  a  maximum  when 

*-*• 

(45)  Find  the  flow  through   a  sewer  consisting  of  a  cast-iron  pipe 
12  inches  diameter,  and  having  a  fall  of  3  feet  per  mile,  when  discharging 
full  bore.     c  =  100. 

(46)  A  pipe  9  inches  diameter  and  one  mile  long  slopes  for  the  first 
half  mile  at  1  in  200  and  for  the  second  half  mile  at  1  in  100.   The  pressure 
head  at  the  higher  end  is  found  to  be  40  feet  of  water  and  at  the  lower 
20  feet. 

Find  the  velocity  and  flow  through  the  pipe. 

Draw  the  hydraulic  gradient  and  find  the  pressure  in  feet  at  500  feet 
and  1000  feet  from  the  higher  end. 

(47)  A  town  of  250,000  inhabitants  is  to  be  supplied  with  water.     Half 
the  daily  supply  of  32  gallons  per  head  is  to  be  delivered  in  8  hours. 

The  service  reservoir  is  two  miles  from  the  town,  and  a  fall  of  10  feet 
per  mile  can  be  allowed  in  the  pipe. 

What  must  be  the  size  of  the  pipe?     C  =  90. 


FLOW  THROUGH   PIPES  175 

(48)  A  water  pipe  is  to  be  laid  in  a  street  800  yards  long  with  houses 
on  both  sides  of  the  street  of  24  feet  frontage.     The  average  number  of 
inhabitants  of  each  house  is  6,  and  the  average  consumption  of  water  for 
each  person  is  30  gallons  in  8  hrs.     On  the  assumption  that  the  pipe  is  laid 
in  four  equal  lengths  of  200  yards  and  has  a  uniform  slope  of  j^y,  and  that 
the  whole  of  the  water  flows  through  the  first  length,  three-fourths  through 
the  second,  one  half  through  the  third  and  one  quarter  through  the  fourth, 
and  that  the  value  of  C  is  90  for  the  whole  pipe,  find  the  diameters  of  the 
four  parts  of  the  pipe. 

(49)  A  pipe  3  miles  long  has  a  uniform  slope  of  20  feet  per  mile,  and  is 
18  inches  diameter  for  the  first  mile,  30  inches  for  the  second  and  21 
inches  for  the  third.     The  pressure  heads  at  the  higher  and  lower  ends  of 
the  pipe  are  100  feet  and  40  feet  respectively.     Find  the  discharge  through 
the  pipe  and  determine  the  pressure  heads  at  the  commencement  of  the 
30  inches  diameter  pipe,  and  also  of  the  21  inches  diameter  pipe. 

(50)  The  difference  of  level  of  two  reservoirs  ten  miles  apart  is  80  feet. 
A  pipe  is  required  to  connect  them  and  to  convey  45,000  gallons  of  water 
per  hour  from  the  higher  to  the  lower  reservoir. 

Find  the  necessary  diameter  of  the  pipe,  and  sketch  the  hydraulic 
gradient,  assuming /=0'01. 

The  middle  part  of  the  pipe  is  120  feet  below  the  surface  of  the  upper 
reservoir.  Calculate  the  pressure  head  in  the  pipe  at  a  point  midway 
between  the  two  reservoirs. 

(51)  Some  hydraulic  machines  are  served  with  water  under  pressure 
by  a  pipe  1000  feet  long,  the  pressure  at  the  machines  being  600  Ibs.  per 
square  inch.     The  horse-power  developed  by  the  machine  is  300  and  the 
friction  horse-power  in  the  pipes  120.     Find  the  necessary  diameter  of  the 

I      v2 

pipe,  taking  the  loss  of  head  in  feet  as  0'03  -3  x  =-  and  '43  Ib.  per  square 

a     2g 

inch  as  equivalent  to  1  foot  head.     Also  determine  the  pressure  at  which 
the  water  is  delivered  by  the  pump. 

What  is  the  maximum  horse-power  at  which  it  would  be  possible  to 
work  the  machines,  the  pump  pressure  remaining  the  same  ?  Lond.  Un. 
1906. 

(52)  Discuss  Reynolds'  work  on  the  critical  velocity  and  on  a  general 
law  of  resistance,  describing  the  experimental  apparatus,  and  showing  the 
connection  with  the  experiments  of  Poiseuille  and  D'Arcy.    Lond.  Un. 
1906. 

(53)  In  a  condenser,  the  water  enters  through  a  pipe  (section  A)  at  the 
bottom  of  the  lower  water  head,  passes  through  the  lower  nest  of  tubes, 
then  through  the  upper  nest  of  tubes  into  the  upper  water  head  (section  B). 
The  sectional  areas  at  sections  A  and  B  are  0'196  and  0'95  sq.  ft.  respec- 
tively ;  the  total  sectional  area  of  flow  of  the  tubes  forming  the  lower  nest 
is  0*814  sq.  ft.,  and  of  the  upper  nest  0'75  sq.  ft.,  the  number  of  tubes  being 
respectively  353  and  326.     The  length  of  all  the  tubes  is  6  feet  2  inches. 
When  the  volume  of  the  circulating  water  was  1*21  c.  ft.  per  sec.,  the 
observed  difference  of  pressure  head  (by  gauges)  at  A  and  B  was  6'5  feet. 
Find  the  total  actual  head  necessary  to  overcome  frictional  resistance,  and 


176  HYDRAULICS 

the  coefficient  of  hydraulic  resistance  referred  to  A.  If  the  coefficient  of 
friction  (4/)  for  the  tubes  is  taken  to  be  '015,  find  the  coefficient  of  hydraulic 
resistance  for  the  tubes  alone,  and  compare  with  the  actual  experiment. 
Lond.  Un.  1906.  (Cr  =  head  lost  divided  by  vel.  head  at  A.) 

(54)  An   open  stream,  which  is  discharging  20  c.  ft.  of  water  per 
second  is  passed  under  a  road  by  a  siphon  of  smooth  stoneware  pipe,  the 
section  of  the  siphon  being  cylindrical,  and  2  feet  in  diameter.     When  the 
stream  enters  this  siphon,  the  siphon    descends  vertically  12    feet,  it 
then  has  a  horizontal  length  of  100  feet,  and  again  rises  12  feet.     If  all  the 
bends  are  sharp  right-angled  bends,  what  is  the  total  loss  of  head  in  the 
tunnel  due  to  the  bends  and  to  the  friction?     C  =  117.    Lond.  Un.  1907. 

(55)  It  has  been  shown  on  page  159  that  when  the  kinetic  energy  of  a 
jet  issuing  from  a  nozzle  on  a  long  pipe  line  is  a  maximum, 


Hence  find  the  minimum  diameter  of  a  pipe  that  will  supply  a  Pelton 
Wheel  of  70  per  cent,  efficiency  and  500  brake  horse-power,  the  available 
head  being  600  feet  and  the  length  of  pipe  3  miles. 

(56)  A  fire  engine  supplies  water  at  a  pressure  of  40  Ibs.  per  square 
inch  by  gauge,  and  at  a  velocity  of  6  feet  per  second  into  a  pipe  3  inches 
diameter.     The  pipe  is  led  a  distance  of  100  feet  to  a  nozzle  25  feet  above 
the  pump.   If  the  coefficient/  (of  friction)  in  the  pipe  be  "01,  and  the  actual 
lift  of  the  jet  is  f  of  that  due  to  the  velocity  of  efflux,  find  the  actual  height 
to  which  the  jet  will  rise,  and  the  diameter  of  the  nozzle  to  satisfy  the 
conditions  of  the  problem. 

(57)  Obtain  expressions  (a)  for  the  head  lost  by  friction  (expressed  in 
feet  of  gas)  in  a  main  of  given  diameter,  when  the  main  is  horizontal,  and 
when  the  variations  of  pressure  are  not  great  enough  to  cause  any  important 
change  of  volume,  and  (6)  for  the  discharge  in  cubic  feet  per  second. 

Apply  your  results  to  the  following  example  :  — 

The  main  is  16  inches  diameter,  the  length  of  the  main  is  300  yards, 
the  density  of  the  gas  is  0'56  (that  of  air  =  l),  and  the  difference  of  pressure 
at  the  two  ends  of  the  pipe  is  ^  inch  of  water  ;  find  :  — 
(a)     The  head  lost  in  feet  of  gas. 
(&)     The  discharge  of  gas  per  hour  in  cubic  feet. 

Weight  of  1  cubic  foot  of  air  =  0'08  lb.;  weight  of  1  cubic  foot  of  water 
62-4  Ibs.  ;  coefficient  /  (of  friction)  for  the  gas  against  the  walls  of  the  pipe 
0-005.  Lond.  Un.  1905. 

(See  page  118  ;  substitute  for  w  the  weight  of  cubic  foot  of  gas.) 

(58)  Three  reservoirs   A,  B   and  C  are  connected  by  a  pipe  leading 
from  each  to  a  junction  box  P  situated  450'  above  datum. 

The  lengths  of  the  pipes  are  respectively  10,000',  5000'  and  6000'  and  the 
levels  of  the  still  water  surface  in  A,  B  and  C  are  800',  600'  and  200'  above 
datum. 

Calculate  the  magnitude  and  indicate  the  direction  of  mean  velocity  in 
each  pipe,  taking  v  =  100  VW,  the  pipes  being  all  the  same  diameter, 
namely  15".  Lond.  Un.  1905. 


FLOW  THROUGH   PIPES  177 

(59)  A  pipe  3'  6"  diameter  bends  through  45  degrees  on  a  radius  of 
25  feet.  Determine  the  displacing  force  in  the  direction  of  the  radial  line 
bisecting  the  angle  between  the  two  limbs  of  the  pipe,  when  the  head  of 
water  in  the  pipe  is  250  feet. 

Show  also  that,  if  a  uniformly  distributed  pressure  be  applied  in  the 
plane  of  the  centre  lines  of  the  pipe,  normally  to  the  pipe  on  its  outer 
surface,  and  of  intensity 

49M2    „ 


per  unit  length,  the  bend  is  in  equilibrium. 
R  =  radius  of  bend  in  feet. 
d=  diameter  of  pipe. 
7&  =  head  of  water  in  the  pipe. 


L.  n. 


12 


CHAPTER  VI. 

FLOW  IN   OPEN  CHANNELS. 

117.  Variety  of  the  forms  of  channels. 

The  study  of  the  flow  of  water  in  open  channels  is  much  more 
complicated  than  in  the  case  of  closed  pipes,  because  of  the 
infinite  variety  of  the  forms  of  the  channels  and  of  the  different 
degrees  of  roughness  of  the  wetted  surfaces,  varying,  as  they  do, 
from  channels  lined  with  smooth  boards  or  cement,  to  the  irregular 
beds  of  rivers  and  the  rough,  pebble  or  rock  strewn,  mountain 
stream. 

Attempts  have  been  made  to  find  formulae  which  are  applicable 
to  any  one  of  these  very  variable  conditions,  but  as  in  the  case  of 
pipes,  the  logarithmic  formulae  vary  with  the  roughness  of  the 
pipe,  so  in  this  case  the  formulae  for  smooth  regular  shaped  channels 
cannot  with  any  degree  of  assurance  be  applied  to  the  calculation 
of  the  flow  in  the  irregular  natural  streams. 

118.  Steady  motion  in  uniform  channels. 

The  experimental  study  of  the  distribution  of  velocities  of 
water  flowing  in  open  channels  reveals  the  fact  that,  as  in  the 
case  of  pipes,  the  particles  of  water  at  different  points  in  a  cross 
section  of  the  stream  may  have  very  different  velocities,  and  the 
direction  of  flow  is  not  always  actually  in  the  direction  of  the  flow 
of  the  stream. 

The  particles  of  water  have  a  sinuous  motion,  and  at  any  point 
it  is  probable  that  the  condition  of  flow  is  continually  changing. 
In  a  channel  of  uniform  section  and  slope,  and  in  which  the  total 
flow  remains  constant  for  an  appreciable  time,  since  the  same 
quantity  of  water  passes  each  section,  the  mean  velocity  v  in  the 
direction  of  the  stream  is  constant,  and  is  the  same  for  all  the 
sections,  and  is  simply  equal  to  the  discharge  divided  by  the  area 
of  the  cross  section.  This  mean  velocity  is  purely  an  artificial 
quantity,  and  does  not  represent,  either  in  direction  or  magnitude, 
the  velocity  of  the  particles  of  water  as  they  pass  the  section. 


FLOW  IN   OPEN  CHANNELS 


179 


Experiments  with,  current  meters,  to  determine  the  distribution 
of  velocity  in  channels,  show,  however,  that  at  any  point  in  the 
cross  section,  the  component  of  velocity  in  a  direction  parallel  to 
the  direction  of  flow  remains  practically  constant.  The  considera- 
tion of  the  motion  is  consequently  simplified  by  assuming  that 
the  water  moves  in  parallel  fillets  or  stream  lines,  the  velocities  in 
which  are  different,  but  the  velocity  in  each  stream  line  remains 
constant.  This  is  the  assumption  that  is  made  in  investigating 
so-called  rational  formulae  for  the  velocity  of  flow  in  channels, 
but  it  should  not  be  overlooked  that  the  actual  motion  may  be 
much,  more  complicated. 

119.  Formula  for  the  flow  when  the  motion  is  uniform 
in  a  channel  of  uniform  section  and  slope. 

On  this  assumption,  the  conditions  of  flow  at  similarly  situated 
points  C  and  D  in  any  two  cross  sections  AA  and  BB,  Figs.  112 
and  113,  of  a  channel  of  uniform  slope  and  section  are  exactly  the 
same ;  the  velocities  are  equal,  and  since  C  and  D  are  at  the  same 
distance  below  the  free  surface  the  pressures  are  also  equal.  For 
the  filament  CD,  therefore, 

PC  +  vJ  =  PD  +  V 
w      2g      w      2g' 

and  therefore,  since  the  same  is  true  for  any  other  filament, 


is  constant  for  the  two  sections. 


Fig.  112. 


Let  v  be  the  mean  velocity  of  the  stream,  i  the  fall  per  foot 
length  of  the  surface  of  the  water,  or  the  slope,  dZ  the  length 
between  AA  and  BB,  ^  the  cross  sectional  area  EFGH  of  the 
stream,  P  the  wetted  perimeter,  i.e.  the  length  EF  +  FG  +  GrH, 
and  w  the  weight  of  a  cubic  foot  of  water. 

Let  ^  =  m  be  called  the  hydraulic  mean  depth. 

Let  dz  be  the  fall  of  the  surface  between  AA  and  BB.  Since 
the  slope  is  small  dz  =  i.dl. 

12—2 


180  HYDRAULICS 

If  Q  cubic  feet  per  second  fall  from  AA  to  BB,  the  work  done 
upon  it  by  gravity  will  be : 

wQ.dz  =  w  .  <*> .  v  .  i  .  dZ. 


Then,  since  2    ^  +  ~ 

\w     lg> 

is  constant  for  the  two  sections,  the  work  done  by  gravity  must 
be  equal  to  the  work  done  by  the  frictional  and  other  resistances 
opposing  the  motion  of  the  water. 

As  remarked  above,  all  the  filaments  have  not  the  same  velocity, 
so  that  there  is  relative  motion  between  consecutive  filaments, 
and  since  water  is  not  a  perfect  fluid  some  portion  of  the  work 
done  by  gravity  is  utilised  in  overcoming  the  friction  due  to  this 
relative  motion.  Energy  is  also  lost,  due  to  the  cross  currents  or 
eddy  motions,  which  are  neglected  in  assuming  stream  line  flow, 
and  some  resistance  is  also  offered  to  the  flow  by  the  air  on  the 
surface  of  the  water. 

The  principal  cause  of  loss  is,  however,  the  frictional  resistance 
of  the  sides  of  the  channel,  and  it  is  assumed  that  the  whole  of 
work  done  by  gravity  is  utilised  in  overcoming  this  resistance. 

Let  F .  v  be  the  work  done  per  unit  area  of  the  sides  of  the 
channel,  v  being  the  mean  velocity  of  flow.  F  is  often  called  the 
frictional  resistance  per  unit  area,  but  this  assumes  that  the  relative 
velocity  of  the  water  and  the  sides  of  the  channel  is  equal  to  the 
mean  velocity,  which  is  not  correct. 

The  area  of  the  surface  of  the  channel  between  AA  and  BB 


Then, 

o>  .     F 

thereiore  ^r  i  =  — 

P       w 

.     F 

or  mi  =  —  . 

w 


F  is  found  by  experiment  to  be  a  function  of  the  velocity  and 
also  of  the  hydraulic  mean  depth,  and  may  be  written 


6  being  a  numerical  coefficient. 

Since  for  water  w  is  constant  —  may  be  replaced  by  k  and 

therefore,  mi  —  If  .  f  (v)  f  (m)  . 

The  form  of  f(v)  f(m)  must  be  determined  by  experiment. 

120.     Formula  of  Chezy. 

The  first  attempts  to  determine  the  flow  of  water  in  channels 


FLOW  IN   OPEN   CHANNELS  181 

with  precision  were  probably  those  of  Chezy  made  on  an  earthen 
canal,  at  Coupalet  in  1775,  from  which  he  concluded  that 
*=«  an 


and  therefore  mi  =  Jcv2  ..............................  (1). 

Writing  C  for  4= 


which  is  known  as  the  Chezy  formula,  and  has  already  been  given 
in  the  chapter  on  pipes. 

121.     Formulae  of  Prony  and  Eytelwein. 
Prony  adopted  the  same  formula  for  channels  and  for  pipes,  and 
assumed  that  F  was  a  function  of  v  and  also  of  v2,  and  therefore, 

mi  =  av  +  bv2. 

By  an  examination  of  the  experiments  of  Chezy  and  those  of 
Du  Buat*  made  in  1782  on  wooden  channels,  20  inches  wide  and 
less  than  1  foot  deep,  and  others  on  the  Jard  canal  and  the  river 
Hayne,  Prony  gave  to  a  and  b  the  values 

a  =  '000044, 
b  =  '000094. 
This  formula  may  be  written 


or 


The  coefficient  C  of  the  Chezy  formula  is  then,  according  to  Prony, 
a  function  of  the  velocity  v. 

If  the  first  term  containing  v  be  neglected,  the  formula  is  the 
same  as  that  of  Chezy,  or 

v  =  103  •Jmi. 

Eytelwein  by  a  re-examination  of  the  same  experiments 
together  with  others  on  the  flow  in  the  rivers  Rhinet  and  Wesert, 
gave  values  to  a  and  b  of 

a  =  '000024, 

b  =  -0001114. 

Neglecting  the  term  containing  a, 

v  =  95  vmJ. 

*  Principes  d'hydraulique. 

t  Experiments  by  Funk,  1803-6. 

J  Experiments  by  Brauings,  1790-92 


182  HYDRAULICS 

As  in  the  case  of  pipes,  Prony  and  Eytelwein  incorrectly 
assumed  that  the  constants  a  and  b  were  independent  of  the 
nature  of  the  bed  of  the  channel. 

122.     Formula  of  Darcy  and  Bazin. 

After  completing  his  classical  experiments  on  flow  in  pipes 
M.  Darcy  commenced  a  series  of  experiments  upon  open  channels 
—  afterwards  completed  by  M.  Bazin  —  to  determine,  how  the 
frictional  resistances  varied  with  the  material  with  which  the 
channels  were  lined  and  also  with  the  form  of  the  channel. 

Experimental  channels  of  semicircular  and  rectangular  section 
were  constructed  at  Dijon,  and  lined  with  different  materials. 
Experiments  were  also  made  upon  the  flow  in  small  earthen 
channels  (branches  of  the  Burgoyne  canal),  earthen  channels  lined 
with  stones,  and  similar  channels  the  beds  of  which  were  covered 
with  mud  and  aquatic  herbs.  The  results  of  these  experiments, 
published  in  1858  in  the  monumental  work,  Recherches  Hydrau- 
liques,  very  clearly  demonstrated  the  inaccuracy  of  the  assump- 
tions of  the  old  writers,  that  the  frictional  resistances  were 
independent  of  the  nature  of  the  wetted  surface. 

From  the  results  of  these  experiments  M.  Bazin  proposed  for 
the  coefficient  fc,  section  120,  the  form  used  by  Darcy  for  pipes, 


a  and  ft  being  coefficients  both  of  which  depend  upon  the  nature 
of  the  lining  of  the  channel. 

Thus,  mi  =  (  a.  +  —  }v2 

\       mj 


or  v  =       ,        =  \mi. 


The  coefficient  C  in  the  Chezy  formula  is  thus  made  to  vary 
with  the  hydraulic  mean  depth  m,  as  well  as  with  the  roughness 
of  the  surface. 

It  is  convenient  to  write  the  coefficient  k  as 


Taking  the  unit  as  1  foot,  Bazin's  values  for  a  and  /?,  and 
values  of  Jc  are  shown  in  Table  XVIII. 

It  will  be  seen  that  the  influence  of  the  second  term  increases 
very  considerably  with  the  roughness  of  the  surface. 

123.     Granguillet  and  Kutter,  from  an  examination  of  Bazin' s 


FLOW   IN   OPEN   CHANNELS 


183 


experiments,  together  with  some  of  their   own,  found   that  the 
coefficient  C  in  the  Chezy  formula  could  be  written  in  the  form 

c=    A       b 


in  which  a  is  a  constant  for  all  channels,  and  b  is  a  coefficient  of 
roughness. 

TABLE   XVIII. 

Showing  the  values  of   a,  ft,  and   k  in  Bazin's  formula  for 
channels. 


a 

ft 

k 

Planed  boards  and  smooth 
cement 

•0000457 

•0000045 

•0000457  I  1  4-         "A 

Rough    boards,   bricks    and 
concrete 

•0000580 

•0000133 

•000058    (lH  -) 
V        m  J 

Ashlar  masonry 

•0000730 

•00006 

(•QO\ 
1  +  —) 
m  / 

Earth 

•0000854 

•00035 

/       4-1  \ 
•0000854(1  +  —) 

\       m  / 

Gravel       (Ganguillet       and 
Kutter) 

•0001219 

•00070 

•0001219(^1  +  —^ 
V        m  J 

The  results  of  experiments  by  Humphreys  and  Abbott  upon 
the  flow  in  the  Mississippi*  were,  however,  found  to  give  results 
inconsistent  with  this  formula  and  also  that  of  Bazin. 

They  then  proposed  to  make  the  coefficient  depend  upon  the 
slope  of  the  channel  as  well  as  upon  the  hydraulic  mean  depth. 

From  experiments  which  they  conducted  in  Switzerland,  upon 
the  flow  in  rough  channels  of  considerable  slope,  and  from  an 
examination  of  the  experiments  of  Humphreys  and  Abbott  on  the 
flow  in  the  Mississippi,  in  which  the  slope  is  very  small,  and 
a  large  number  of  experiments  on  channels  of  intermediate  slopes, 
they  gave  to  the  coefficient  C,  the  unit  being  1  foot,  the  value 

0-00281     1-811 


41'6-t- 


c  = 


n 


in  which  n  is  a  coefficient  of  roughness  of  the  channel  and  has  the 
values  given  in  Tables  XIX  and  XIX  A. 


*  Report  on  the  Hydraulics  of  the  Mississippi  River,  1861 ;  Flow  of  water  in 
rivers  and  canals,  Trautwine  and  Bering,  1893. 


184  HYDKAULICS 

TABLE  XIX. 

Showing  values  of  n  in  the  formula  of  Granguillet  and  Kutter. 

Channel  n 

Yery  smooth,  cement  and  planed  boards        '009  to  '01 

Smooth,  boards,  bricks,  concrete          '012  to '013 

Smooth,  covered  with  slime  or  tuberculated '015 

Rough  ashlar  or  rubble  masonry          '017  to '019 

Very  firm  gravel  or  pitched  with  stones         ...         ...         ...  '02 

Earth,  in  ordinary  condition  free  from  stones  and  weeds  ...  '025 

Earth,  not  free  from  stones  and  weeds  -030 

Gravel  in  bad  condition '035  to '040 

Torrential  streams  with  rough  stony  beds     -05 

TABLE  XIX  A. 

^howing  values  of  n  in  the  formula  of  Granguillet  and  Kutter, 

determined  from  recent  experiments. 

n 

Rectangular  wooden  flume,  very  smooth       '0098 

Wood  pipe  6  ft.  diameter  -0132 

Brick,  washed  with  cement,  basket  shaped  sewer,  6'  x  6'  8",  nearly 

new -0130 

Brick,  washed  with  cement,  basket  shaped  sewer,  6'x6'8",  one 

year  old         " -0148 

Brick,  washed  with  cement,  basket  shaped  sewer,  6'x6'8",  four 

years  old       '0152 

Brick,  washed  with  cement,  circular  sewer,  9  ft.  diameter,  nearly 

new -0116 

Brick,  washed  with  cement,  circular  sewer,  9  ft.  diameter,  four 

years  old       -0133 

Old  Croton  aqueduct,  lined  with  brick  '015 

New  Croton  aqueduct* -012 

Sudbury  aqueduct  ...         ...         ...         ...         ...         ...         ...  '01 

Glasgow  aqueduct,  lined  with  cement  ...         ...         ...         ...  '0124 

Steel  pipe,  wetted,  clean,  1897  (mean)  -0144 

Steel  pipe,  1899  (mean) -0155 

This  formula  has  found  favour  with  English,  American  and 
German  engineers,  but  French  writers  favour  the  simpler  formula 
of  Bazin. 

It  is  a  peculiarity  of  the  formula,  that  when  m  equals  unity 

then  C  =  -  and  is  independent  of  the  slope ;  and  also  when  m  is 

large,  C  increases  as  the  slope  decreases. 

It  is  also  of  importance  to  notice  that  later  experiments  upon 
the  Mississippi  by  a  special  commission,  and  others  on  the  flow  of 
the  Irrawaddi  and  various  European  rivers,  are  inconsistent  with 

*  Report  New  York  Aqueduct  Commission. 


FLOW   IN   OPEN   CHANNELS  185 

the  early  experiments  of  Humphreys  and  Abbott,  to  which 
G-anguillet  and  Kutter  attached  very  considerable  importance  in 
framing  their  formula,  and  the  later  experiments  show,  as  described 
later,  that  the  experimental  determination  of  the  flow  in,  and  the 
slope  of,  large  natural  streams  is  beset  with  such  great  difficulties, 
that  any  formula  deduced  for  channels  of  uniform  section  and 
slope  cannot  with  confidence  be  applied  to  natural  streams,  and 
vice  versa. 

The  application  of  this  formula  to  the  calculation  of  uniform 
channels  gives,  however,  excellent  results,  and  providing  the  value 
of  n  is  known,  it  can  be  used  with  confidence. 

It  is,  however,  very  cumbersome,  and  does  not  appear  to  give 
results  more  accurate  than  a  new  and  simpler  formula  suggested 
recently  by  Bazin  and  which  is  given  in  the  next  section. 

124.     M.  Bazin's  later  formula  for  the  flow  in  channels. 

M.  Bazin  has  recently  (Annales  des  Ponts  et  Chaussees,  1897, 
Vol.  IV.  p.  20),  made  a  careful  examination  of  practically  all  the 
available  experiments  upon  channels,  and  has  proposed  for  the 
coefficient  C  in  the  Chezy  formula  a  form  originally  proposed  by 
G-anguillet  and  Kutter,  which  he  writes 

1 


or  0  = 


in  which  a  is  constant  for  all  channels  and  ft  is  a  coefficient  of 
roughness  of  the  channel. 

O 

Taking  1  metre  as  the  unit  a  =  "0115,  and  writing  y  for  -, 

(1), 

i ,  y_ 

*Jm 

or  when  the  unit  is  one  foot, 


1+4= 


(2), 


the  value  of  y  in  (2)  being  l'811y,  in  formula  (1). 

The  values  of  y  as  found  by  Bazin  for  various  kinds  of  channels 
are  shown  in  Table  XX,  and  in  Table  XXI  are  shown  values  of 


186  HYDRAULICS 

C,   to    the    nearest    whole    number,    as    deduced    from    Bazin's 
coefficients  for  values  of  m  from  '2  to  50. 

For  the  channels  in  the  first  four  columns  only  a  very  few 
experimental  values  for  C  have  been  obtained  for  values  of  m 
greater  than  3,  and  none  for  m  greater  than  7'3.  For  the  earth 
channels,  experimental  values  for  C  are  wanting  for  small  values 
of  m,  so  that  the  values  as  given  in  the  table  when  m  is  greater 
than  7 '3  for  the  first  four  columns,  and  those  for  the  first  three 
columns  for  m  less  than  1,  are  obtained  on  the  assumption,  that 
Bazin's  formula  is  true  for  all  values  of  m  within  the  limits  of  the 
table. 

TABLE  XX. 

Values  of  y  in  the  formula, 

o= 157-5 


unit  metre    unit  foot 

Very  smooth  surfaces  of  cement  and  planed  boards  ...  '06  -1085 

Smooth  surfaces  of  boards,  bricks,  concrete '16  '29 

Ashlar  or  rubble  masonry  "46  '83 

Earthen  channels,  very  regular  or  pitched  with  stones, 

tunnels  and  canals  in  rock '85  1'54 

Earthen  channels  in  ordinary  condition  T30  2'35 

Earthern  channels  presenting  an  exceptional  resistance, 
the  wetted  surface  being  covered  with  detritus, 

stones  or  weed,  or  very  irregular  rocky  surface  1*7  3'17 

125.     Glazed  earthenware  pipes. 

Vellut*  from  experiments  on  the  flow  in  earthenware  pipes  has 
given  to  C  the  value 


in  which 
or 


This  gives  values  of  C,  not  very  different  from  those  given  by 
Bazin's  formula  when  y  is  0'29. 

In  Table  XXI,  column  2,  glazed  earthenware  pipes  have  been 
included  with  the  linings  given  by  Bazin. 

*  Proc.  I.  C.  E.,  Vol.  CLI.  p.  482. 


FLOW   IN   OPEN  CHANNELS 


187 


TABLE  XXI. 

Values  of  C  in  the  formula  v  =  C\/wn  calculated  from  Bazin's 
formula,  the  unit  of  length  being  1  foot, 

c=  157'5 


Channels 

Hydraulic 
mean 
depth 

Very  smooth 
cement  and 
planed 
boards 

Smooth 
boards,  brick, 
concrete, 
glazed 
earthenware 

Smooth 
but  dirty 
brick, 
concrete 

Ashlar 
masonry 

Earth  canals 
in  very  good 
condition, 
and  canals 
pitched  with 

Earth  canals 
in  ordinary 
condition 

Earth  canals 
exceptionally 
rough 

m. 

pipes 

stones 

y  =  '1085 

y  =  '29 

7  =  -50 

y  =  '83 

7  =  1-54 

7  =  2-35 

y  =  3-17 

•2 

127 

96 

74 

55 

35 

25 

19 

•3 

131 

103 

82 

63 

41 

30 

23 

•4 

135 

108 

88 

68 

46 

32 

26 

•5 

137 

112 

92 

72 

50 

37 

29 

•6 

139 

116 

96 

76 

53 

39 

31 

•8 

141 

119 

101 

82 

58 

43 

35 

1-0 

142 

122 

105 

86 

62 

47 

38 

1-3 

144 

126 

109 

91 

67 

51 

42 

1-5 

145 

128 

112 

94 

70 

54 

44 

1-75 

146 

130 

114 

97 

73 

57 

46 

2-0 

147 

132 

116 

99 

76 

59 

49 

2-5 

148 

134 

119 

103 

80 

64 

53 

3-0 

149 

136 

122 

107 

84 

67 

56 

4-0 

150 

138 

126 

111 

89 

72 

61 

5-0 

151 

140 

129 

115 

94 

77 

65 

6-0 

151 

142 

131 

118 

98 

80 

69 

8-0 

152 

144 

134 

122 

102 

86 

74 

10-0 

153 

145 

136 

125 

106 

90 

79 

12-0 

109 

94 

82 

15-0 

113 

98 

87 

20-0 

117 

103 

92 

30-0 

123 

110 

100 

50-0 

129 

119 

108 

126.    Bazin's  method  of  determining  a  and  ft. 
The  method  used  by  Bazin  to  determine  the  values  of  a  and  ft 
is  of  sufficient  interest  and  importance  to  be  considered  in  detail. 


He  first  calculated  values  of  -7=  and 

Vm 


from  experimental 


data,  and  plotted   these  values  as   shown  in   Fig.   114,   -j= 

abscissae,  and  -    '-  as  ordinates. 
v 


as 


188 


HYDRAULICS 


As  will  be  seen  on  reference  to  the  figure,  points  have  been 
plotted  for  four  classes  of  channels,  and  the  points  lie  close  to  four 
straight  lines  passing  through  a  common  point  P  on  the  axis 
of  y. 


The  equation  to  each  of  these  lines  is 


FLOW   IN   OPEN   CHANNELS  189 

'Jmi  8 

or  -  =  a  +  -£  - 

^  vm 

a  being  the  intercept  on  the  axis  of  y,  or  the  ordinate  when  -f=  is 

v/m 
zero,  and  /?,  which  is  variable,  is  the  inclination  of  any  one  of 

these  lines  to  the  axis  o 
transposing  the  equation, 


these  lines  to  the  axis  of  x  ;  for  when  —  =  is  zero,  ^-      =  a,  and 

vm 


\l 


which  is  clearly  the  tangent  of  the  angle  of  inclination  of  the  line 
to  the  axis  of  x. 

It  should  be  noted,  that  since  -  =  ~  ,  the  ordinates  give 

actual  experimental  values  of  g  ,  or  by  inverting  the  scale,  values 

of  C.     Two  scales  for  ordinates  are  thus  shown. 

In  addition  to  the  points  shown  on  the  diagram,  Fig.  114, 
Bazin  plotted  the  results  of  some  hundreds  of  experiments  for  all 
kinds  of  channels,  and  found  that  the  points  lay  about  a  series  of 
lines,  all  passing  through  the  point  P,  Fig.  114,  for  which  a  is  '00635, 

and  the  values  of  -  ,  i.e.  y,  are  as  shown  in  Table  XX. 

CL 

Bazin  therefore  concluded,  that  for  all  channels 

^'=  -00635  +*, 

v  Vm 

the  value  of  /?  depending  upon  the  roughness  of  the  channel. 

For  very  smooth  channels  in  cement  and  planed  boards,  Bazin 
plotted  a  large  number  of  points,  not  shown  in  Fig.  114,  and  the 
line  for  which  7  =  '109  passes  very  nearly  through  the  centre  of 
the  zone  occupied  by  these  points. 

The  line  for  which  y  is  0'29  coincides  well  with  the  mean  of 
the  plotted  points  for  smooth  channels,  but  for  some  of  the  points 
y  may  be  as  high  as  0'4. 

It  is  further  of  interest  to  notice,  that  where  the  surfaces  and 
sections  of  the  channels  are  as  nearly  as  possible  of  the  same 
character,  as  for  instance  in  the  Boston  and  New  York  aqueducts, 
the  values  of  the  coefficient  C  differ  by  about  6  per  cent.,  the 
difference  being  probably  due  to  the  pointing  of  the  sides  and 
arch  of  the  New  York  aqueduct  not  being  so  carefully  executed 
as  for  the  Boston  aqueduct.  By  simply  washing  the  walls  of  the 
latter  with  cement,  Fteley  found  that  its  discharge  was  increased 
20  per  cent. 


190  HYDRAULICS 

y  is  also  greater  for  rectangular-shaped  channels,  or  those 
which  approximate  to  the  rectangular  form,  than  for  those  of 
circular  form,  as  is  seen  by  comparing  the  two  channels  in  wood 
W  and  P,  and  also  the  circular  and  basket-shaped  sewers. 

M.  Bazin  also  found  that  y  was  slightly  greater  for  a  very 
smooth  rectangular  channel  lined  with  cement  than  for  one  of 
semicircular  section. 

In  the  figure  the  author  has  also  plotted  the  results  of  some 
recent  experiments,  which  show  clearly  the  effect  of  slime  and 
tuberculations,  in  increasing  the  resistance  of  very  smooth  channels. 
The  value  of  y  for  the  basket-shaped  sewer  lined  with  brick, 
washed  with  cement,  rising  from  *4  to  '642  during  4  years'  service. 

127.     Variations  in  the  coefficient  C. 

For  channels  lined  with  rubble,  or  similar  materials,  some  of 
the  experimental  points  give  values  of  C  differing  very  considerably 
from  those  given  by  points  on  the  line  for  which  y  is  0*83,  Fig.  114, 
but  the  values  of  C  deduced  from  experiments  on  particular 
channels  show  similar  discrepancies  among  themselves. 

On  reference  to  Bazin's  original  paper  it  will  be  seen  that,  for 
channels  in  earth,  there  is  a  still  greater  variation  between  the 
experimental  values  of  C,  and  those  given  by  the  formula,  but  the 
experimental  results  in  these  cases,  for  any  given  channel,  are 
even  more  inconsistent  amongst  themselves. 

An  apparently  more  serious  difficulty  arises  with  respect  to 
Bazin's  formula  in  that  C  cannot  be  greater  than  157'5.  The 
maximum  value  of  the  hydraulic  mean  depth  m  recorded  in 
any  series  of  experiments  is  74*3,  obtained  by  Humphreys  and 
Abbott  from  measurements  of  the  Mississippi  at  Carrollton  in  1851. 
Taking  y  as  2'35  the  maximum  value  for  C  would  then  be  124. 
Humphreys  and  Abbott  deduced  from  their  experiments  values 
of  C  as  large  as  254.  If,  therefore,  the  experiments  are  reliable 
the  formula  of  Bazin  evidently  gives  inaccurate  results  for  excep- 
tional values  of  m. 

The  values  of  C  obtained  at  Carrollton  are,  however,  incon- 
sistent with  those  obtained  by  the  same  workers  at  Vicksburg, 
and  they  are  not  confirmed  by  later  experiments  carried  out  at 
Carrollton  by  the  Mississippi  commission.  Further  the  velocities 
at  Carrollton  were  obtained  by  double  floats,  and,  according  to 
G-ordon*,  the  apparent  velocities  determined  by  such  floats  should 
be  at  least  increased,  when  the  depth  of  the  water  is  large,  by  ten 
per  cent. 

Bazin  has  applied  this  correction  to  the  velocities  obtained  by 

*  Gordon,  Proceedings  Inst.  Civil  Eng.,  1893. 


FLOW    IN    OPEN    CHANNELS  191 

Humphreys  and  Abbott  at  Vicksburg  and  also  to  those  obtained 
by  the  Mississippi  Commission  at  Carrollton,  and  shows,  that  the 
maximum  value  for  C  is  then,  probably,  only  122. 

That  the  values  of  C  as  deduced  from  the  early  experiments  on 
the  Mississippi  are  unreliable,  is  more  than  probable,  since  the 
smallest  slope,  as  measured,  was  only  '0000034,  which  is  less  than 
^  inch  per  mile.  It  is  almost  impossible  to  believe  that  such  small 
differences  of  level  could  be  measured  with  certainty,  as  the 
smallest  ripple  would  mean  a  very  large  percentage  error,  and 
it  is  further  probable  that  the  local  variations  in  level  would  be 
greater  than  this  measured  difference  for  a  mile  length.  Further, 
assuming  the  slope  is  correct,  it  seems  probable  that  the  velocity 
under  such  a  fall  would  be  less  than  some  critical  velocity  similar 
to  that  obtained  in  pipes,  and  that  the  velocity  instead  of  being 
proportional  to  the  square  root  of  the  slope  i,  is  proportional 
to  i.  That  either  the  measured  slope  was  unreliable,  or  that  the 
velocity  was  less  than  the  critical  velocity,  seems  certain  from  the 
fact,  that  experiments  at  other  parts  of  the  Mississippi,  upon  the 
Irrawaddi  by  Gordon,  and  upon  the  large  rivers  of  Europe,  in  no 
case  give  values  of  C  greater  than  124. 

The  experimental  evidence  for  these  natural  streams  tends, 
however,  clearly  to  show,  that  the  formulae,  which  can  with 
confidence  be  applied  to  the  calculation  of  flow  in  channels  of 
definite  form,  cannot  with  assurance  be  used  to  determine  the 
discharge  of  rivers.  The  reason  for  this  is  not  far  to  seek,  as 
the  conditions  obtaining  in  a  river  bed  are  generally  very  far 
removed  from  those  assumed,  in  obtaining  the  formula.  The 
assumption  that  the  motion  is  uniform  over  a  length  sufficiently 
great  to  be  able  to  measure  with  precision  the  fall  of  the  surface, 
must  be  far  from  the  truth  in  the  case  of  rivers,  as  the  irregu- 
larities in  the  cross  section  must  cause  a  corresponding  variation 
in  the  mean  velocities  in  those  sections. 

In  the  derivation  of  the  formula,  frictional  resistances  only 
are  taken  into  account,  whereas  a  considerable  amount  of  the 
work  done  on  the  falling  water  by  gravity  is  probably  dissipated 
by  eddy  motions,  set  up  as  the  stream  encounters  obstructions  in 
the  bed  of  the  river.  These  eddy  motions  must  depend  very 
much  on  local  circumstances  and  will  be  much  more  serious  in 
irregular  channels  and  those  strewn  with  weeds,  stones  or  other 
obstructions,  than  in  the  regular  channels.  Another  and  probably 
more  serious  difficulty  is  the  assumption  that  the  slope  is  uniform 
throughout  the  whole  length  over  which  it  is  measured,  whereas 
the  slope  between  two  cross  sections  may  vary  considerably 
between  bank  and  bank.  It  is  also  doubtful  whether  locally 


192  HYDRAULICS 

there  is  always  equilibrium  between  the  resisting  and  accelerating 
forces.  In  those  cases,  therefore,  in  which  the  beds  are  rocky  or 
covered  with  weeds,  or  in  which  the  stream  has  a  very  irregular 
shape,  the  hypotheses  of  uniform  motion,  slope,  and  section,  will 
not  even  be  approximately  realised. 

128.    Logarithmic  formula  for  the  flow  in  channels. 

In  the  formulae  discussed,  it  has  been  assumed  that  the  f  rictional 
resistance  of  the  channel  varies  as  the  square  of  the  velocity,  and 
in  order  to  make  the  formulae  fit  the  experiments,  the  coefficient  C 
has  been  made  to  vary  with  the  velocity. 

As  early  as  1816,  Du  Buat*  pointed  out,  that  the  slope  i 
increased  at  a  less  rate  than  the  square  of  the  velocity,  and 
half  a  century  later  St  Tenant  proposed  the  formula 

m*=  -000401  «tt. 

To  determine  the  discharge  of  brick-lined  sewers,  Mr  Santo 
Crimp  has  suggested  the  formula 

v  =  124m°'6V5 

and  experiments  show  that  for  sewers  that  have  been  in  use  some 
time  it  gives  good  results.  The  formula  may  be  written 


An  examination  of  the  results  of  experiments,  by  logarithmic 
plotting,  shows  that  in  any  uniform  channel  the  slope 

.  -w; 

~  mp> 

~k  being  a  numerical  coefficient  which  depends  upon  the  roughness 
of  the  surface  of  the  channel,  and  n  and  p  also  vary  with  the 
nature  of  the  surface. 

Therefore,  in  the  formula, 


From  what  follows  it  will  be  seen  that  n  varies  between  1'75 
and  2'1,  while  p  varies  between  1  and  1'5. 

kvn 
Since  m  is  constant,  the  formula  i=—j,  may  be  written  i  =  bvn, 

b  being  equal  to  — -p . 

Therefore  log  i  =  log  b  +  n  log  v. 

*  Principes  d'Hydraulique,  Vol.  i.  p.  29,  1816. 


FLOW   IN   OPEN   CHANNELS 


193 


In  Fig.  115  are  shown  plotted  the  logarithms  of  i  and  v 
obtained  from  an  experiment  by  Bazin  on  the  flow  in  a  semi- 
circular cement-lined  pipe.  The  points  lie  about  a  straight  line, 
the  tangent  of  the  inclination  of  which  to  the  axis  of  v  is  1'96 
and  the  intercept  on  the  axis  of  i  through  v  =  1,  or  log  v  =  0,  is 
•0000808. 


Fig.  115.    Logarithmic  plottings  of  i  and  v  to  determine  the  index  n  in 

kvn 

the  formula  for  channels,  t=—  . 
m*> 


For  this  experimental  channel,  therefore, 


In  the  same  figure  are  shown  the  plottings  of  log  i  and  log  v  for 
the  siphon-aqueduct*  of  St  Elvo  lined  with  brick  and  for  which 
m  is  278  feet.  In  this  case  n  is  2  and  b  is  '000283.  Therefore 

i=  '000283^. 

If,  therefore,  values  of  v  and  i  are  determined  for  a  channel, 
while  m  is  kept  constant,  n  can  be  found. 


*  Annales  des  Fonts  et  Cliaussees,  Vol.  iv.  1897. 


L.   H. 


13 


194  HYDRAULICS 


To  determine  the  ratio  - .    The  formula, 
P 


m 
may  be  written  in  the  form, 


i 

v,     n 


or  om  =  o-     +- 

By  determining  experimentally  m  and  vy  while  the  slope  i  is 
kept  constant,  and  plotting  log  m  as  ordinates  and  log  v  as 
abscissae,  the  plottings  lie  about  a  straight  line,  the  tangent  of  the 

*Y1 

inclination   of  which  to   the   axis  of  v  is  equal  to   -,  and  the 
intercept  on  the  axis  of  m  is  equal  to 


In  Fig.  116  are  shown  the  logarithmic  plottings  of  m  and  v  for 
a  number  of  channels,  of  varying  degrees  of  roughness. 

m 

The  ratio  -  varies  considerably,  and  for  very  regular  channels 

increases  with  the  roughness  of  the  channel,  being  about  1*40  for 
very  smooth  channels,  lined  with  pure  cement,  planed  wood  or 
cement  mixed  with  very  fine  sand,  1*54  for  channels  in  unplaned 
wood,  and  1*635  for  channels  lined  with  hard  brick,  smooth 
concrete,  or  brick  washed  with  cement.  For  channels  of  greater 

roughness,  -  is  very  variable  and  appears  to  become  nearly  equal 
to  or  even  less  than  its  value  for  smooth  channels.  Only  in  one 
case  does  the  ratio  -  become  equal  to  2,  and  the  values  of  m  and 

v  for  that  case  are  of  very  doubtful  accuracy. 

As  shown  above,  from  experiments  in  which  m  is  kept  constant, 

n  can  be  determined,  and  since  by  keeping  i  constant  -  can  be 

found,  n  and  p  can  be  deduced  from  two  sets  of  experiments. 

Unfortunately,  there  are  wanting  experiments  in  which  m  is 
kept  constant,  so  that,  except  for  a  very  few  cases,  n  cannot 
directly  be  determined. 

There  is,  however,  a  considerable  amount  of  experimental  data 
for  channels  similarly  lined,  and  of  different  slopes,  but  here 


FLOW  IN   OPEN   CHANNELS 


195 


Fig.  116.     Logarithmic  plottings  of  m  and  v  to  determine  the 


n  .  ,     kv 

ratio  -  in  the  formula  i  =  — -  . 


TABLE  XXII. 
Particulars  of  channels,  plottings  for  which  are  shown  in  Fig.  116. 


1.  Semicircular  channel,  very  smooth,  lined  with  wood 

2.  ,,  ,,  ,,  ,,  ,,         ,,     cement     mixed    with 
very  fine  sand 

3.  Eectangular  channel,  very  smooth,  lined  with  cement  

4.  ,          ,,  ,,  ,      wood,  1'  1"  wide 

5.  ,  smooth  ,          „      slope  -00208   ... 

6.  ,  ,    „    „   '0043  ... 

7.  ,  ,    „    „   -0049  ... 

8.  ,  ,    „    „   -00824  ... 

9.  New  Croton  aqueduct,  smooth,  lined  with  bricks  (Keport  New  York 

Water  Supply) 

10.  Glasgow  aqueduct,  smooth,  lined  with  concrete  (Proc.  I.  C.  E.  1896) 

11.  Sudbury        „  ,,  ,,         ,,     brick  well  pointed  (Tr.  Am. 

S.C.E.  1883) 

12.  Boston  sewer,  circular,  smooth,  lined  with  brick  washed  with  cement 

(Tr.  Am.S.  G.  E.  1901)          

13.  Eectangular  channel,  smooth,  lined  with  brick 

14.  „  „  ,,  „         „     wood 

15.  ,,  ,,  ,,  ,,         ,,     small  pebbles 
15a.  Kectangular  sluice  channel  lined  with  hammered  ashlar 


16.  ,,  channel  lined  with  large  pebbles  ... 

17.  Torlonia  tunnel,  rock,  partly  lined 

18.  Ordinary  channel  lined  with  stones  covered  with  mud  and  weeds 

19-  „             „          „         „         „           „           „       „       „       „ 

20.  Eiver  Weser  

21.  „         „  

22.  „         „  

23.  Earth  channel.     Gros  bois 

24.  Cavour  canal 

25.  Eiver  Seine  ... 


n 

P 

1-45 

1-36 
1-44 
1-38 
1-54 
1-54 
1-54 
1-54 

1-74 
1-635 

1-635 

1-635 
1-635 
1-655 
1-49 
1-36 
1-36 
1-29 
1-49 
1-18 
•94 
1-615 
1-65 
2-1 
1-49 
1-5 
1-37 


13—2 


196 


HYDRAULICS 


again,  as  will  appear  in  the  context,  a  difficulty  is  encountered,  as 
even  with  similarly  lined  channels,  the  roughness  is  in  no  two 
cases  exactly  the  same,  and  as  shown  by  the  plottings  in  Fig.  116, 
no  two  channels  of  any  class  give  exactly  the  same  values 


n 


for  - ,  but  for  certain  classes  the  ratio  is  fairly  constant. 

Taking,  for  example,  the  wooden  channels  of  the  group  (Nos.  4 


n 


to  8),  the  values  of  —  are  all  nearly  equal  to  1*54. 

The  plottings  for  these  channels  are  again  shown  in  Fig.  117. 
The  intercepts  on  the  axis  of  m  vary  from  0'043  to  0'14. 


09 
-08 

•07 

-06 
-OS 


e 


4.    Leg  v 


Fig.  117.     Logarithmic  plottings  to  determine  the  ratio  -  for  smooth  channels. 

Let  the  intercepts  on  the  axis  of  m  be  denoted  by  y,  then, 

i 


y  = 


FLOW   IN   OPEN   CHANNELS 


197 


and 


logy=-logfc--log*. 


If  k  and  p  are  constant  for  these  channels,  and  logi  and 
log  y  are  plotted  as  abscissae  and  ordinates,  the  plottings  should  lie 
about  a  straight  line,  the  tangent  of  the  inclination  of  which  to  the 

axis  of  i  is  -  ,  and  when  log  y  =  0,  or  y  is  unity,  the  abscissa  i  =  k, 

i.e.  the  intercept  on  the  axis  of  i  is  k. 

In  Fig.  118  are  shown  the  plottings  of  log  i  and  log  y  for  these 
channels,  from  which  p=l'14>  approximately,  and  k  =  '00023. 

Therefore,  n  is  approximately  1'76,  and  taking  -  as  1'54 

.     •00023i;176 
*~      m1'14      ' 


•01 


•OU5 


— 


t 


•OOO2 


'-OOO23 


-0005. 


\ 


N; 


-OO1 


-O02 


-005 


Log.  is 


Fig.  118.     Logarithmic  plottings  to  determine  the  value  of  p  for  smooth 
channels,  in  the  formula  i  —  —  -. 


Since  the  ratio  -  is  not  exactly  1'54  for  all  these  channels,  the 

P 

values  of  n  and  p  cannot  be  exactly  correct  for  the  four  channels, 
but,  as  will  be  seen  on  reference  to  Table  XXIII,  in  which  are 
shown  values  of  v  as  observed  and  as  calculated  by  the  formula, 
the  calculated  and  observed  values  of  v  agree  very  nearly. 


198 


HYDRAULICS 


TABLE  XXIII. 

Values  of  v,  for  rectangular   channels  lined  with  wood,  as 
determined  experimentally,  and  as  calculated  from  the  formula 

ii1'76 
i  =  -00028  £™. 


Slope  -00208 

Slope  -0049 

Slope  -00824 

v  ob- 

v calcu- 

v ob- 

v calcu- 

v ob- 

v calcu- 

m in 

served 

lated 

m  in 

served 

lated 

m  in 

served 

lated 

metres 

metres 

metres 

metres 

metres 

metres 

metres 

metres 

metres 

per  sec. 

per  sec. 

per  sec. 

per  sec. 

per  sec. 

per  sec. 

0-1381 

0-962 

0-972 

0-1042 

1-325 

1-314 

•0882 

1-594 

1-589 

•1609 

1-076 

1-07 

•1224 

1-479 

1-459 

•1041 

1-776 

1-764 

•1832 

1-152 

1-165 

•1382 

1-612 

1-58 

•1197 

1-902 

1-932 

•1976 

1-259  ' 

1-223 

•1535 

1-711 

1-690 

•1313 

2-053 

2-051 

•2146 

1-324 

1-290 

•1668 

1-818 

1-782 

•1420 

2-186 

2-158 

•2313 

1-374 

1-354 

•1789 

1-898 

1-858 

•1543 

2-268 

2-275 

•2441 

1-440 

1-402 

•1913 

1-967 

1-947 

•1649 

2-357 

2-377 

•2578 

1-487 

1-452 

•2018 

2-045 

2-014 

•1744 

2-447 

2-460 

•2681 

1-552 

1-49 

•2129 

2-102 

2-089 

•1842 

2-518 

2-553 

•2809 

1-587 

1-552 

•2215 

2-179 

2-143 

•1919 

2-612 

2-618 

As  a  further  example,  which  also  shows  how  n  and  p  increase 
with  the  roughness  of  the  channel,  consider  two  channels  built  in 
hammered  ashlar,  for  which  the  logarithmic  plottings  of  m  and  v 

are  shown  in  Fig.  116,  Nos.  15  a  and  156,  and  —  is  1'36. 

The  slopes  of  these  channels  are  '101  and  '037.  By  plotting 
log^  and  logy,  p  is  found  to  be  1'43  and  k  '000149.  So  that  for 
these  two  channels 

•000149t?r95 


The  calculated  and  observed  velocities  are  shown  in  Table  XXXI 
and  agree  remarkably  well. 


Very  smooth  channels. 


The  ratio  -  for  the  four  very  smooth 


channels,  shown  in  Fig.  116,  varies  between  1'36  and  1'45,  the 
average  value  being  about  1'4.  On  plotting  logy  and  logi  the 
points  did  not  appear  to  lie  about  any  particular  line,  so  that  p 
could  not  be  determined,  and  indicates  that  k  is  different  for  the 
four  channels.  Trial  values  of  n  =  175  and  p  =  1*25  were  taken,  or 


and  values  of  k  calculated  for  each  channel. 


FLOW   IN   OPEN   CHANNELS  199 

Velocities  as  determined  experimentally  and  as  calculated  for 
three  of  the  channels  are  shown  in  Table  XXIII  from  which  it  will 
be  seen  that  k  varies  from  '00006516  for  the  channel  lined  with 
pure  cement,  to  '0001072  for  the  rectangular  shaped  section  lined 
with  carefully  planed  boards. 

It  will  be  seen,  that  although  the  range  of  velocities  is  con- 
siderable, there  is  a  remarkable  agreement  between  the  calculated 
and  observed  values  of  v,  so  that  for  very  smooth  channels  the 
values  of  n  and  p  taken,  can  be  used  with  considerable  confidence. 

Channels  moderately  smooth.  The  plottings  of  log  m  and  log  v 
for  channels  lined  with  brick,  concrete,  and  brick  washed  with 
cement  are  shown  in  Fig.  116,  Nos.  9  to  13. 

It  will  be  seen  that  the  value  of  -  is  not  so  constant  as  for  the 

P 
two  classes  previously  considered,  but  the  mean  value  is  about 

nj 

1*635,  which  is  exactly  the  value  of  -  for  the  Sudbury  aqueduct. 

*¥! 

For  the  New  Croton  aqueduct  -  is  as  high  as  1'74,  and,  as  shown 
in  Fig.  114,  this  aqueduct  is  a  little  rougher  than  the  Sudbury. 

The  variable  values   of  --   show   that  for  any  two  of  these 

P 

channels  either  n,  or  p,  or  both,  are  different.  On  plotting  logi 
and  log?;  as  was  done  in  Fig.  115,  the  points,  as  in  the  last  case, 
could  not  be  said  to  lie  about  any  particular  straight  line,  and  the 
value  of  p  is  therefore  uncertain.  It  was  assumed  to  be  1'15,  and 

*¥l 

therefore,  taking  -  as  1'635,  n  is  1*88. 

Since  no  two  channels  have  the  same  value  for  -  ,  it  is  to  be 

P 
expected  that  the  coefficient  Jc  will  not  be  constant. 

In  the  Tables  XXIV  to  XXXIII  the  values  of  v  as  observed 
and  as  calculated  from  the  formula 

.     kv™ 


and  also  the  value  of  k  are  given. 

It  will  be  seen  that  Jc  varies  very  considerably,  but,  for  the 
three  large  aqueducts  which  were  built  with  care,  it  is  fairly 
constant. 

The  effect  of  the  sides  of  the  channel  becoming  dirty  with 
time,  is  very  well  seen  in  the  case  of  the  circular  and  basket- 
shaped  sewers.  In  the  one  case  the  value  of  &,  during  four  years' 
service,  varied  from  '00006124  to  '00007998  and  in  the  other  from 
'00008405  to  '0001096.  It  is  further  of  interest  to  note,  that  when 


200  HYDRAULICS 

m  and  v  are  both  unity  and  k  is  equal  to  "000067,  the  value  of  i  is 
the  same  as  given  by  Bazin's  formula,  when  y  is  '29,  and  when  Jc  is 
"0001096,  as  in  the  case  of  the  dirty  basket-shaped  sewer,  the  value 
of  y  is  '642,  which  agrees  with  that  shown  for  this  sewer  on 
Fig.  114. 

Channels  in  masonry.  Hammered  ashlar  and  rubble.  Attention 
has  already  been  called,  page  198,  to  the  results  given  in 
Table  XXXI  for  the  two  channels  lined  with  hammered  ashlar. 

The  values  of  n  and  p  for  these  two  channels  were  determined 
directly  from  the  logarithmic  plottings,  but  the  data  is  insufficient 
to  give  definite  values,  in  general,  to  n,  p,  and  k. 

In  addition  to  these  two  channels,  the  results  for  one  of 
Bazin's  channels  lined  with  small  pebbles,  and  for  other  channels 
lined  with  rubble  masonry  and  large  pebbles  are  given.  The 

ratio  -  is  quoted  at  the  head  of  the  tables  where  possible.     In  the 
p 

other  cases  n  and  p  were  determined  by  trial. 

The  value  of  n,  for  these  rough  channels,  approximates  to  2, 
and  appears  to  have  a  mean  value  of  about  1'96,  while  p  varies 
from  1'36  to  1'5. 

Earthen  channels.  A  very  large  number  of  experiments  have 
been  made  on  the  flow  in  canals  and  rivers,  but  as  it  is  generally 

impracticable  to  keep  either  i  or  m  constant,  the  ratio  -  can  only 

be  determined  in  a  very  few  cases,  and  in  these,  as  will  be  seen 
from  the  plottings  in  Fig.  116,  the  results  are  not  satisfactory,  and 

'I? 

appear  to  be  unreliable,  as  -  varies  between  "94  and  2'18.    It  seems 

probable  that  p  is  between  1  and  1*5  and  n  from  1*96  to  2'15. 
Logarithmic  formulae  for  various  classes  of  channels. 
Very  smooth  channels,  lined  with  cement,  or  planed  boards, 


i=  ('000065  to  '00011)  —    . 
lined  with  brick  we 
*  =  '000065  to  '00011 


Smooth  channels,  lined  with  brick  well  pointed,  or  concrete, 


mx" 
Channels  lined  with  ashlar  masonry,  or  small  pebbles, 

,1-96 

*« -00015^. 

m14 

Channels  lined  with  rubble  masonry,  large  pebbles,  rock,  and 
exceptionally  smooth  earth  channels  free  from  deposits, 

i;1'96 
t  =  '00028 -^n-,. 

Ill 


FLOW   IN   OPEN   CHANNELS  201 

Earth  channels, 


w1*' 

k  varies  from  '00033  to  '00050  for  channels  in  ordinary  condition 
and  from  '00050  to  "00085  for  channels  of  exceptional  resistance. 

129.  Approximate  formula  for  the  flow  in  earth 
channels. 

The  author  has  by  trial  found  n  and  p  for  a  number  of 
channels,  and  except  for  very  rough  channels,  n  is  not  very 
different  from  2,  and  p  is  nearly  1'5.  The  approximate  formula 

v  =  C  v  m%  i, 

may,  therefore,  be  taken  for  earth  channels,  in  which  C  is  about 
50  for  channels  in  ordinary  condition. 

In  Table  XXXIII  are  shown  values  of  v  as  observed  and 
calculated  from  this  formula. 

The  hydraulic  mean  depth  varies  from  '958  to  14*1  and  for  all 
values  between  these  external  limits,  the  calculated  velocities 
agree  with  the  observed,  within  10  per  cent.,  whereas  the  variation 
of  C  in  the  ordinary  Chezy  formula  is  from  40  to  103,  and 
according  to  Bazin's  formula,  C  would  vary  from  about  60  to  115. 
With  this  formula  velocities  can  be  readily  calculated  with  the 
ordinary  slide  rule. 

TABLE  XXIY. 
Yery  smooth  channels. 
Planed  wood,  rectangular,  1'575  wide. 

»  =  -0001072^, 

II  V 

log  k  =  4'0300. 

v  ft.  per  sec.     v  ft.  per  sec. 
m  feet  observed         calculated 

•2372  3-55  3'57 

•2811  4-00  4-03 

•3044  4-20  4-26 

•3468  4-67  4'68 

•3717  4-94  4-94 

•3930  5-11  5-12 

•4124  5-26  5-30 

•4311  5-49  5-47 


202  HYDRAULICS 

TABLE  XXIV  (continued). 
Pure  cement,  semicircular. 


ra 

«"• 

•00006516  ^5, 

log  fc  =  5-8141. 

m  t;  observed      v  calculated 

•503  3-72  3-66 

•682  4-59  4-55 

•750  4-87  4-87 

•915  5-57  5-62 

1-034  6-14  6-14 

Cement  and  very  fine  sand,  semicircular. 
i  =  '0000759^, 
log  fc  =  5-8802. 


m  feet 

•379 
•529 
•636 
•706 
•787 
•839 
•900 
•941 
•983 

1-006 

1-02 

1-04 


TABLE  XXV. 

Boston  circular  sewer,  9  ft.  diameter. 

Brick,  washed  with  cement,  t  =  srnn7  (Horton). 

i  =  -00006124^, 
m115 

log  t?  =  -61 18  log  m  +  '5319  logt  +  2*2401, 

v  ft.  per  sec.     v  ft.  per  sec. 
m  feet  observed          calculated 

•928  2-21  2-34 

1-208  2-70  2-76 

1-408  3-03  3-03 

1-830  3-48  3-56 

1-999  3-73  3-75 

2-309  4-18  4-10 


v  ft.  per  sec. 

v  ft.  per  sec. 

observed 

calculated 

2-87 

2-74 

3-44 

3-49 

3-87 

3-98 

4-30 

4-30 

4-51 

4-59 

4-80 

4-84 

4-94 

5-10 

5-20 

5-26 

5-38 

5-43 

5-48 

5-53 

5-55 

5-58 

5-66 

5-66 

FLOW   IN   OPEN   CHANNELS  203 

TABLE  XXV    (continued). 
The  same  sewer  after  4  years'  service. 

i  =  '00007998^, 
m115 

log  v  =  '6118  log  m+  -5319  log  i  +  21795. 

m  v  observed  v  calculated 
1-120                2-38  2-29 

1-606  2-82  2-78 

1-952  3-16  3-22 

2-130  3-30  3-39 

TABLE  XXYI. 

New  Croton  aqueduct.     Lined  with  concrete. 

i  =  -000073^, 
logv  =  '61 18  log  m  +  '5319  log  *+  2'200. 


v  ft.  per  sec. 

v  ft.  per  sec. 

m  feet 

observed 

calculated 

1-000 

1-37 

1-37 

1-250 

1-59 

1-57 

1-499 

1-79 

1-76 

1-748 

1-95 

1-93 

2-001 

2-11 

2-10 

2-250 

2-27 

2-26 

2-500 

2-41 

2-40 

2-749 

2-52 

2-55 

2-998 

2-65 

2-68 

3-251 

2-78 

2-82 

3-508 

2-89 

2-96 

3-750 

3-00 

3-08 

3-838 

3-02 

3-12 

TABLE  XXVII. 

Sudbury  aqueduct.     Lined  with  well  pointed  brick. 

i  =  -00006427  ^ps, 
m115 

log  &  =  5-808  log  v  =  -6118  log  m  +  '5319  log  i  +  2'23. 

v  ft.  per  sec.     v  ft.  per  sec. 

m  feet  observed          calculated 

•4987  1-135  1-142 

•6004  1-269  1-279 

•8005  1-515  1-525 

1-000  1-755  1-752 

1-200  1-948  1-954 

1-400  2-149  2-147 

1-601  2-332  2-331 

1-801  2-513  2-511 

2-001  2-651  2-672 

2-201  2-844  2-832 

2-336  2-929  2-937 


204  HYDRAULICS 

TABLE  XXVIII. 
Rectangular  channel  lined  with,  brick  (Bazin). 

4,1-88 

i=  '000107^. 
m11" 

v  ft.  per  sec.     v  ft.  per  sec. 
m  feet  observed          calculated 

•1922  2-75  2-90 

•2838  3-67  3'68 

•3654  4-18  4-30 

•4235  4-72  4-71 

•4812  5-10  5-09 

•540  5-34  5-46 

•5823  5-68  5'77 

•6197  6-01  5-94 

•6682  6-15  6-22 

•6968  6-47  6'39 

•7388  6-60  6-62 

•7788  6-72  6'83 

G-lasgow  aqueduct.     Lined  with  concrete. 

i  = '0000696^, 

m115 

log  v  =  "6118  log  m  +  -5319  log  i  +  2'2113. 

v  ft.  per  sec.     v  ft.  per  sec. 
m  feet  observed          calculated 

1-227  1-87  1'89 

1-473  2-07  2-11 

1-473  2-106  2-11 

1-489  2-214  2-13 

1-499  2-13  2-14 

1-499  2-15  2-14 

1-548  2-18  2-22 

1-597  2-21  2-23 

1-607  2-23  2-23 

1-610  2-22  2-24 

1-620  2-24  2-24 

1-627  2-25  2-27 

1-738  2-26  2-33 

1-811  2-47  2-40 

TABLE  XXIX. 

Charlestown  basket-shaped  sewer  6'  x  6'  8". 
Brick,  washed  with  cement,  *'  =  21^577  (Horton). 

i= -00008405^, 
m115 

log  v  =  '6118  log  m  +  "5319  log  i  +  2'1678. 

v  ft.  per  sec.     v  ft.  per  sec. 
m  feet  observed          calculated 

•688  1-99  2-05 

•958  2-46  2-52 

1-187  2-82  2-87 

1-539  3-44  3-36 


FLOW    IN   OPEN    CHANNELS 


205 


TABLE  XXIX   (continued). 

The  same  sewer  after  4  years'  service, 

-.1-88 

;  = -0001096  ~-5, 
log  v  =  -6118  log  m  +  -5319  log  i  +  2'1065. 


m  feet 
1-342 
1-508 
1-645 


v  ft.  per  sec.     v  ft.  per  sec. 
observed          calculated 


2-66 
2-86 
3-04 


2-68 
2-88 
3-04 


TABLE  XXX. 

Left  aqueduct  of  the  Solani  canal,  rectangular  in  section,  lined 
with  rubble  masonry  (Cunningham), 


i 

•000225 
•000206 
•000222 
•000207 
•000189  ? 


V96 

i=  -00026-^. 
m14 

v  ft.  per  sec.     v  ft.  per  sec. 
m  feet          observed          calculated 


6-43 

6-81 

7-21 

7-643 

7-94 


3-46 
3-49 
3-70 

3-87 
4-06 


Right  aqueduct, 


=  -0002213  ~, 
m14 


•000195 
•000225 
•000205 
•000193 
•000193 
•000190 


m 

3-42 
5-86 
6-76 
7-43 
7-77 
7-96 


v  observed 
2-43 
3-61 
3-73 
3-87 
3-93 
4-06 


3-50 
3-47 
3-84 
3-83 
3-83 


v  calculated 
2-26 
3-58 
3-76 
3-89 
4-04 
4-06 


Torlonia  tunnel,  partly  in  hammered  ashlar,  partly  in  solid 
rock, 

i=  '00104, 

,»1  "95 

i  =  '00022 


TOJ 


1-932 
2-172 
2-552 
2-696 
3-251 
3-438 
3-531 
3-718 


v  observed 

v  calculated 

3-382 

3-45 

3-625 

3-73 

4-232 

4-16 

4-324 

4-32 

5-046 

4-90 

4-965 

5-08 

4-908 

5-18 

5-358 

5-37 

206 


HYDRAULICS 


TABLE  XXXI. 

Channel  lined  with  hammered  ashlar, 

P~ 

1-96 

t =-000149  ^~ 


log  fc  =  41740. 


-"-—> 

v  ft.  per  sec. 

v  ft.  per  sec. 

m  feet 

observed 

calculated 

•324 

12-30 

12-30 

•467 

16-18 

16-18 

•580 

18-68 

18-97 

•562 

21-09 

20-8 

m  feet 

•424 
•620 

•745 
•852 


ft.  per  sec.    v  ft.  per  sec. 
observed          calculated 


9-04 
11-46 
13-55 
15-08 


9-02 
11-86 
13-52 
14-93 


Channel  lined  with  small  pebbles,  ^  =  '0049  (n=l'96, 
will  give  equally  good  results). 

%l-4», 

p 

1*95 

t» -000152^= 


log  fc  =  41913. 


m  feet 

•250 
•357 
•450 
•520 
•588 
•644 
•700 
•746 
•785 
•832 
•871 
•910 


v  ft.  per  sec. 
observed 

2-16 
2-95 
3-40 
3-84 
4-14 
4-43 
4-64 
4-88 
5-12 
5-26 
5-43 
5-57 


v  ft.  per  sec. 
calculated 

2-34 
2-97 
3-47 
3-82 
4-15 
4-43 
4-66 
4-88 
5-05 
5-25 
5-43 
5-58 


FLOW   IN    OPEN    CHANNELS 


207 


TABLE  XXXII. 

Channel  lined  with  large  pebbles  (Bazin), 

-.1-96 

i  =  '000229^, 
log  Je  -4-3605. 

v  ft.  per  sec.     v  ft.  per  sec. 
m  feet  observed        calculated 


•291 
•417 
•510 
•587 
•656 
•712 
•772 
•823 
•867 
•909 
•946 
•987 


1-79 
2-43 
2-90 
3-27 
3-56 
3-85 
4-03 
4-23 
4-43 
4-60 
4-78 
4-90 


1-84 
2-44 
2-90 
3-18 
3-45 
3-67 
3-91 
4-33 
4-53 
4-69 
4-84 
5-00 


TABLE  XXXIII. 

Velocities  as  observed,  and  as  calculated  by  the  formula 
v=cJ^$~i.    0  =  50. 

Ganges  Canal. 


•000155 
•000229 
•000174 
•000227 
•000291 


m  feet 

5-40 
8-69 
7-82 
9-34 
4-50 


v  ft.  per  sec.    v  ft.  per  sec. 
observed          calculated 


2-4 
3-71 
2-96 
4-02 

2-82 


2-34 
3-80 
3-08 
4-00 
2-63 


•0005503 
•0005503 
•0002494 
•0002494 


River  Weser. 
m  v  observed 


8-93 
13-35 
14-1 
10'5 


6-29 
7'90 
5-69 
4-75 


v  calculated 

6-0 

8-18 

5-70 

4-78 


•0001183 
•0001782 
•0001714 
•0002130 


Missouri, 
m  v  observed       v  calculated 


10-7 
12-3 
15-4 
17-7 


3-6 
4-38 
5-03 
6-19 


3-23 
4-37 
4-80 
6-26 


208 


•00029 
•00029 
•00033 
•00033 


HYDRAULICS 

Cavour 

Canal. 

m 

v  observed 

7-32 
5-15 
5-63 
4-74 

3-70 
3-10 
3-40 
3-04 

v  calculated 

3-80 
2-92 
3-14 
2-91 


Earth  channel  (branch  of  Burgoyne  canal). 
Some  stones  and  a  few  herbs  upon  the  surface. 

0  =  48. 


•000957 
•000929 
•000993 
•000986 
•000792 
•000808 
•000858 
•000842 


m  feet 

•958 
1-181 
1-405 
1-538 

•958 
1-210 
1-436 
1-558 


v  ft.  per  sec.     v  ft.  per  sec. 
observed         calculated 


1-243 
1-702 
1-797 
1-958 
1-233 
1-666 
1-814 
1-998 


1-30 
1-66 
1-94 
2-06 
1-25 
1-56 
1-79 
2-08 


130.  Distribution  of  the  velocity  in  the  cross  section 
of  open  channels. 

The  mean  velocity  of  flow  in  channels  and  pipes  of  small  cross 
sectional  area  can  be  determined  by  actually  measuring  the  weight 
or  the  volume  of  the  water  discharged,  as  shown  in  Chapter  VII, 
and  dividing  the  volume  discharged  per  second  by  the  cross 
section  of  the  pipe.  For  large  channels  this  is  impossible,  and 
the  mean  velocity  has  to  be  determined  by  other  means,  usually 
by  observing  the  velocity  at  a  large  number  of  points  in  the  same 
transverse  section  by  means  of  floats,  current  meters*,  or  Pi  tot 
tubes  t.  If  the  bed  of  the  stream  is  carefully  sounded,  the  cross 
section  can  be  plotted  and  divided  into  small  areas,  at  the  centres 
of  which  the  velocities  have  been  observed.  If  then,  the  observed 
velocity  be  assumed  equal  to  the  mean  velocity  over  the  small 
area,  the  discharge  is  found  by  adding  the  products  of  the  areas 
and  velocities. 

Or  Q  -  2a .  v. 

M.  Bazin+,  with  a  thoroughness  that  has  characterised  his 
experiments  in  other  branches  of  hydraulics,  has  investigated  the 
distribution  of  velocities  in  experimental  channels  and  also  in 
natural  streams. 

In  Figs.  119  and  120  respectively  are  shown  the  cross  sections 
of  an  open  and  closed  rectangular  channel  with  curves  of  equal 


See  page  238. 

Bazin,  Recherches  Hydraulique, 


t  See  page  241. 


FLOW   IN    OPEN    CHANNELS 


209 


velocity  drawn  on  the  section.  Curves  showing  the  distribution 
of  velocities  at  different  depths  on  vertical  and  horizontal  sections 
are  also  shown. 


relo cities  onJfcrizontaLSectio 


cu- 


Curves  ofeqiLoL  Velocity 

Fig.  119. 


Velocities    on 
Vertical*  Sections. 


/" 

* 

'' 

V\ 

\ 

^ 

/ 

' 

,' 

-r*                                    --K, 

X 

\ 

v^ 

/ 

Velocities  on, 
ttori^oTital  Sections. 

\ 

1 

ab          c     d    e 

a, 

b 

C        A        ^ 

/ 

^- 

4   /?     X     X  /  ^ 

/ 

^ 

el  f/     /    / 

( 

N 

(^                                                                      "N 

"g"!  j-   *            &• 

7-763         1-176 

- 

^'«1    <       v       \ 

^>fe  \\\        x        \          i. 

\ 

\ 

^                                                 J 

_ 

1 

^  \\'\          N        \ 

(L 

*> 

v 

C          tt'           & 

J 

\\    \    \y 

\ 

^. 

y 

/ 

v\i^  Xx-^    ^v     ^- 

Fig.  120. 

It  will  be  seen  that  the  maximum  velocity  does  not  occur  in 
the  free  surface  of  the  water,  but  on  the  central  vertical  section 
at  some  distance  from  the  surface,  and  that  the  surface  velocity 
may  be  very  different  from  the  mean  velocity.  As  the  maximum 
velocity  does  not  occur  at  the  surface,  it  would  appear  that  in 
L.  H.  H 


210  HYDRAULICS 

assuming  the  wetted  perimeter  to  be  only  the  wetted  surface  of 
the  channel,  some  error  is  introduced.  That  the  air  has  not  the 
same  influence  as  if  the  water  were  in  contact  with  a  surface 
similar  to  that  of  the  sides  of  the  channel,  is  very  clearly 
shown  by  comparing  the  curves  of  equal  velocity  for  the  closed 
rectangular  channel  as  shown  in  Fig.  119  with  those  of  Fig.  120. 
The  air  resistance,  no  doubt,  accounts  in  some  measure  for  the 
surface  velocity  not  being  the  maximum  velocity,  but  that  it  does 
not  wholly  account  for  it  is  shown  by  the  fact  that,  whether  the 
wind  is  blowing  up  or  down  stream,  the  maximum  velocity  is  still 
below  the  surface.  M.  Flamant*  suggests  as  the  principal  reason 
why  the  maximum  velocity  does  not  occur  at  the  surface,  that 
the  water  is  less  constrained  at  the  surface,  and  that  irregular 
movements  of  all  kinds  are  set  up,  and  energy  is  therefore 
utilised  in  giving  motions  to  the  water  not  in  the  direction  of 
translation. 

Depth  on  any  vertical  at  which  the  velocity  is  equal  to  the  mean 
velocity.  Later  is  discussed,  in  detail,  the  distribution  of  velocity 
on  the  verticals  of  any  cross  section,  and  it  will  be  seen,  that  if  u 
is  the  mean  velocity  on  any  vertical  section  of  the  channel,  the 
depth  at  which  the  velocity  is  equal  to  the  mean  velocity  is  about 
0'6  of  the  total  depth.  This  depth  varies  with  the  roughness  of 
the  stream,  and  is  deeper  the  greater  the  ratio  of  the  depth  to 
the  width  of  the  stream.  It  varies  between  '5  and  '55  of  the  depth 
for  rivers  of  small  depth,  having  beds  of  fine  sand,  and  from  '55 
to  '66  in  large  rivers  from  1  to  3J  feet  deep  and  having  strong 
bedst. 

As  the  banks  of  the  stream  are  approached,  the  point  at  which 
the  mean  velocity  occurs  falls  nearer  still  to  the  bed  of  the  stream, 
but  if  it  falls  very  low  there  is  generally  a  second  point  near  the 
surface  at  which  the  velocity  is  also  equal  to  the  mean  velocity. 

When  the  river  is  covered  with  ice  the  maximum  velocity  of 
the  current  is  at  a  depth  of  "35  to  '45  of  the  total  depth,  and  the 
mean  velocity  at  two  points  at  depths  of  '08  to  '13  and  '68  to  '74 
of  the  total  depth  {. 

If,  therefore,  on  various  verticals  of  the  cross  section  of  a  stream 
the  velocity  is  determined,  by  means  of  a  current  meter,  or  Pitot 
tube,  at  a  depth  of  about  '6  of  the  total  depth  from  the  surface, 
the  velocity  obtained  may  be  taken  as  the  mean  velocity  upon  the 
vertical. 

*  Hydraulique. 

t  Le  Genie  Civil,  April,  1906,   "Analysis  of  a  communication  by  Murphy  to 
the  Hydrological  section  of  the  Institute  of  Geology  of  the  United  States." 
J  Cunningham,  Experiments  on  the  Ganges  Canal. 


FLOW   IN   OPEN    CHANNELS 


211 


The  total  discharge  can  then  be  found,  approximately,  by 
dividing  the  cross  section  into  a  number  of  rectangles,  such  as 
abed,  Fig.  120  a,  and  multiplying  the  area  of  the  rectangle  by  the 
velocity  measured  on  the  median  line  at  0*6  of  its  depth. 


cu    d 


Fig.  120a. 

The  flow  of  the  Upper  Nile  has  recently  been  determined  in 
this  way. 

Captain  Cunningham  has  given  several  formulae,  for  the  mean 
velocity  u  upon  a  vertical  section,  of  which  two  are  here  quoted. 

(1), 
(2), 

"V  being  the  velocity  at  the  surface,  va  the  velocity  at  f  of  the  depth, 
v±  at  one  quarter  of  the  depth,  and  so  on. 

131.  Form  of  the  curve  of  velocities  on  a  vertical 
section. 

M.  Bazin*  and  Cunningham  have  both  taken  the  curve  of 
velocities  upon  a  vertical  section  as  a  parabola,  the  maximum 
velocity  being  at  some  distance  hm  below  the  free  surface  of  the 
water. 

Let  V  be  the  velocity  measured  at  the  centre  of  a  current  and 
as  near  the  surface  as  possible.  This  point  will  really  be  at  1  inch 
or  more  below  the  surface,  but  it  is  supposed  to  be  at  the  surface. 

Let  v  be  the  velocity  on  the  same  vertical  section  at  any  depth 
h,  and  H  the  depth  of  the  stream. 

Bazin  found  that,  if  the  stream  is  wide  compared  to  its  depth, 
the  relationship  between  v,  V,  h,  and  i  the  slope,  is  expressed  by 
the  formula, 

" 


-  I-    — 

?ffi~*VH 


or 


k  being  a  numerical  coefficient,  which  has  a  nearly  constant  value 
of  36'2  when  the  unit  of  length  is  one  foot. 

*  Recherches  Hydraulique,  p.  228 ;  Annales  des  Fonts  et  Cliaussees,  2nd  Vol., 
1875. 

14—2 


212  HYDRAULICS 

To  determine  the  depth  on  any  vertical  at  which  the  velocity  is 
equal  to  the  mean  velocity.  Let  u  be  the  mean  velocity  on  any 
vertical  section,  and  hu  the  depth  at  which  the  velocity  is  equal  to 
the  mean  velocity. 

The  discharge  through  a  vertical  strip  of  width  9Z  is 

/•H 

u~Kdl=dll    v.dh. 
Jo 

/•H  /  1.7,2      _  .  \ 

Therefore  ^H  =  J0    (V  -  j£  </H3  )«fik> 

and  u  =  V-|VHJ    ..............................  (2). 

Substituting  u  and  hu  in  (1)  and  equating  to  (2), 


H 
and  fc«  =  '577H. 

This  depth,  at  which  the  velocity  is  equal  to  the  mean  velocity, 
is  determined  on  the  assumption  that  fc  is  constant,  which  is  only 
true  for  sections  very  near  to  the  centre  of  streams  which  are 
wide  compared  with  their  depth. 

It  will  be  seen  from  the  curves  of  Fig.  120  that  the  depth  at 
which  the  maximum  velocity  occurs  becomes  greater  as  the  sides. 
of  the  channel  are  approached,  and  the  law  of  variation  of  velocity 
also  becomes  more  complicated.  M.  Bazin  also  found  that  the 
depth  at  the  centre  of  the  stream,  at  which  the  maximum  velocity 
occurs,  depends  upon  the  ratio  of  the  width  to  the  depth,  the 
reason  apparently  being  that,  in  a  stream  which  is  wide  compared 
to  its  depth,  the  flow  at  the  centre  is  but  slightly  affected  by  the 
resistance  of  the  sides,  but  if  the  depth  is  large  compared  with  the 
width,  the  effect  of  the  sides  is  felt  even  at  the  centre  of  the 
stream.  The  farther  the  vertical  section  considered  is  removed 
from  the  centre,  the  effect  of  the  resistance  of  the  sides  is 
increased,  and  the  distribution  of  velocity  is  influenced  to  a 
greater  degree.  This  effect  of  the  sides,  Bazin  expressed  by 
making  the  coefficient  k  to  vary  with  the  depth  hm  at  which 
the  maximum  velocity  occurs. 

The  coefficient  is  then, 

,         36'2 

FIT 

Further,  the  equation  to  the  parabola  can  be  written  in  terms 
of  vm,  the  maximum  velocity,  instead  of  V. 


FLOW   IN   OPEN   CHANNELS  213 

THUS,  .  —  -  M*  «     (*-*->'  ..................  (3). 


The  mean  velocity  u,  upon  the  vertical  section,  is  then, 

u  =  ^pp  /     vdh 
H;  o 

36-2 
=  Vm  — 


H 

Therefore 


H;  u  VA   H 

When  v  —  u,     h  =  hu, 

and  therefore,  "*  "  m . 

o       ±1      ±1  H 

The  depth  7im  at  which  the  velocity  is  a  maximum  is  generally 
less  than  *2H,  except  very  near  the  sides,  and  Tiu  is,  therefore,  not 
very  different  from  *6H,  as  stated  above. 

Ratio  of  maximum,  velocity  to  the  mean  velocity.  From 
equation  (4), 

hnt,          hn?\ 


H, 

In  a  wide  stream  in  which  the  depth  of  a  cross  section  is  fairly 
constant  the  hydraulic  mean  depth  m  does  not  differ  very  much 
from  H,  and  since  the  mean  velocity  of  flow  through  the  section  is 
C  *Jmi  and  is  approximately  equal  to  ut  therefore, 


36-2        /I     hm 


Assuming  7tm  to  vary  from  0  to  "2  and  C  to  be  100,  —  varies 

from   T12   to   1'09.     The   ratio   of    maximum   velocity  to  mean 
velocity  is,  therefore,  probably  not  very  different  from  I'l. 

132.  The  slopes  of  channels  and  the  velocities  allowed 
in  them. 

The  discharge  of  a  channel  being  the  product  of  the  area  and 
the  velocity,  a  given  discharge  can  be  obtained  by  making  the 
area  small  and  the  velocity  great,  or  vice  versa.  And  since  the 
velocity  is  equal  to  CVmi,  a  given  velocity  can  be  obtained  by 


214  HYDRAULICS 

varying  either  m  or  i.  Since  m  will  in  general  increase  with  the 
area,  the  area  will  be  a  minim  urn  when  i  is  as  large  as  possible. 
But,  as  the  cost  of  a  channel,  including  land,  excavation  and 
construction,  will,  in  many  cases,  be  almost  proportional  to  its 
cross  sectional  area,  for  the  first  cost  to  be  small  it  is  desirable 
that  i  should  be  large.  It  should  be  noted,  however,  that  the 
discharge  is  generally  increased  in  a  greater  proportion,  by  an 
increase  in  A,  than  for  the  same  proportional  increase  in  i. 

Assume,  for  instance,  the  channel  to  be  semicircular. 

The  area  is  proportional  to  d2,  and  the  velocity  v  to  *Jd.  i. 

Therefore  Q  oc  d?  4di. 

If  d  is  kept  constant  and  i  doubled,  the  discharge  is  increased 
to  s/2Q,  but  if  d  is  doubled,  i  being  kept  constant,  the  discharge 
will  be  increased  to  5'6Q.  The  maximum  slope  that  can  be  given 
will  in  many  cases  be  determined  by  the  difference  in  level  of  the 
two  points  connected  by  the  channel. 

When  water  is  to  be  conveyed  long  distances,  it  is  often 
necessary  to  have  several  pumping  stations  en  route,  as  sufficient 
fall  cannot  be  obtained  to  admit  of  the  aqueduct  or  pipe  line  being 
laid  in  one  continuous  length. 

The  mean  velocity  in  large  aqueducts  is  about  3  feet  per 
second,  while  the  slopes  vary  from  1  in  2000  to  1  in  10,000.  The 
slope  may  be  as  high  as  1  in  1000,  but  should  not,  only  iri  excep- 
tional circumstances,  be  less  than  1  in  10,000. 

In  Table  XXXIV  are  given  the  slopes  and  the  maximum 
velocities  in  them,  of  a  number  of  brick  and  masonry  lined 
aqueducts  and  earthen  channels,  from  which  it  will  be  seen  that 
the  maximum  velocities  are  between  2  and  5|  feet  per  second, 
and  the  slopes  vary  from  1  in  2000  to  1  in  7700  for  the  brick  and 
masonry  lined  aqueducts,  and  from  1  in  300  to  1  in  20,000  for  the 
earth  channels.  The  slopes  of  large  natural  streams  are  in  some 
cases  even  less  than  1  in  100,000.  If  the  velocity  is  too  small 
suspended  matter  is  deposited  and  slimy  growths  adhere  to  the  sides. 

It  is  desirable  that  the  smallest  velocity  in  the  channel  shall  be 
such,  that  the  channel  is  "  self-cleansing,"  and  as  far  as  possible 
the  growth  of  low  forms  of  plant  life  prevented. 

In  sewers,  or  channels  conveying  unfiltered  waters,  it  is 
especially  desirable  that  the  velocity  shall  not  be  too  small,  and 
should,  if  possible,  not  be  less  than  2  ft.  per  second. 

TABLE  XXXIY. 

Showing  the  slopes  of,  and  maximum  velocities,  as  determined 
experimentally,  in  some  existing  channels. 


FLOW   IN    OPEN   CHANNELS 


215 


Smooth  aqueducts. 


New  Croton  aqueduct 
Sudbury  aqueduct 
Glasgow  aqueduct 
Paris  Dhuis 
Avre,  1st  part 
„       2nd  part 
Manchester  Thirlmere 
Naples 
Boston  Sewer 


Slope 

•0001326 

•000189 

•000182 

•000130 

•0004 

•00033 

•000315 

•00050 

•0005 

•000333 


Maximum  velocity 

3       ft.  per  second 

2-94 

2-25 


4-08 
3-44 

4-18 


Earth  channels. 
Slope  Maximum  velocity 


Ganges  canal 

Escher      „ 

Linth         „ 

Cavour       „ 

Simmen    ,, 

Chazilly  cut 

Marseilles  canal 

Chicago  drainage  canal 

(of  the  bottom  of  the  canal) 


Lining 


•000306 
•003 
•00037 
•00033 
•0070 
•00085 
•00043 

4-16  ft.  p 
4-08 
5-53 
3-42 
3-74 
1-70 
1-70 

er  sec 

ond         earth 

V 

(  gravel  and 
}  some  stones 
earth 
(  earth,  stony, 
(  few  weeds 

•00005 


TABLE  XXXV. 

Showing  for  varying  values  of  the  hydraulic  mean  depth  m,  the 
minimum  slopes,  which  brick  channels  and  glazed  earthenware 
pipes  should  have,  that  the  velocity  may  not  be  less  than  2  ft. 
per  second. 


in  feet 

•1 
•2 
•3 
•4 
•5 
•6 
•8 
1-0 


slope 

1     in     93 
275 
510 

775 
1058 
1380 
2040 


m  feet 

1-25 

1-5 

1-75 

2-0 

2-5 

3-0 

4-0 


slope 

1  in  3700 
4700 
5710 
6675 
9000 
11200 
15850 


2760 


The  slopes  are  calculated  from  the  formula 


The  value  of  y  is  taken  as  0*5  to  allow  for  the  channel  becoming 
dirty.     For  the  minimum  slope  for  any  other  velocity  v,  multiply 

(2\2 
-J  .     For  example,  the  minimum  slope 

for  a  velocity  of  3  feet  per  second  when  m  is  1,  is  1  in  1227. 


216  HYDRAULICS 

Velocity  of  flow  in,  and  slope  of  earth  channels.  If  the  velocity 
is  high  in  earth  channels,  the  sides  and  bed  of  the  channel  are 
eroded,  while  on  the  other  hand  if  it  is  too  small,  the  capacity  of 
the  channel  will  be  rapidly  diminished  by  the  deposition  of  sand 
and  other  suspended  matter,  and  the  growth  of  aquatic  plants. 
Du  Buat  gives  '5  foot  per  second  as  the  minimum  velocity  that 
mud  shall  not  be  deposited,  while  Belgrand  allows  a  minimum 
of  *8  foot  per  second. 

TABLE  XXXVI. 

Showing  the  velocities  above  which,  according  to  Du  Buat, 
and  as  quoted  by  Rankine,  erosion  of  channels  of  various  materials 
takes  place. 

Soft  clay  0'25  ft.  per  second 

Fine  sand  0-50 

Coarse  sand  and  gravel  as  large  as  peas        0'70 

Gravel  1  inch  diameter  2*25 

Pebbles  1|  inches  diameter  3 '33 

Heavy  shingle  4-00 

Soft  rock,  brick,  earthenware  4'50 

Rock,  various  kinds  6'00  and  upwards 

133.  Sections  of  aqueducts  and  sewers. 

The  forms  of  sections  given  to  some  aqueducts  and  sewers  are 
shown  in  Figs.  121  to  131.  In  designing  such  aqueducts  and 
sewers,  consideration  has  to  be  given  to  problems  other  than  the 
comparatively  simple  one  of  determining  the  size  and  slope  to 
be  given  to  the  channel  to  convey  a  certain  quantity  of  water. 
The  nature  of  the  strata  through  which  the  aqueduct  is  to  be 
cut,  and  whether  the  excavation  can  best  be  accomplished  by 
tunnelling,  or  by  cut  and  cover,  and  also,  whether  the  aqueduct 
is  to  be  lined,  or  cut  in  solid  rock,  must  be  considered.  In  many 
cases  it  is  desirable  that  the  aqueduct  or  sewer  should  have  such 
a  form  that  a  man  can  conveniently  walk  along  it,  although  its 
sectional  area  is  not  required  to  be  exceptionally  large.  In 
such  cases  the  section  of  the  channel  is  made  deep  and  narrow. 
For  sewers,  the  oval  section,  Figs.  126  and  127,  is  largely 
adopted  because  of  the  facilities  it  gives  in  this  respect,  and  it  has 
the  further  advantage  that,  as  the  flow  diminishes,  the  cross 
section  also  diminishes,  and  the  velocity  remains  nearly  constant 
for  all,  except  very  small,  discharges.  This  is  important,  as  at 
small  velocities  sediment  tends  to  collect  at  the  bottom  of  the 
sewer. 

134.  Siphons  forming  part  of  aqueducts. 

It  is  frequently  necessary  for  some  part  of  an  aqueduct  to  be 
constructed  as  a  siphon,  as  when  a  valley  has  to  be  crossed  or  the 


FLOW   IN   OPEN   CHANNELS 


217 


aqueduct  taken  under  a  stream  or  other  obstruction,  and  the 
aqueduct  must,  therefore,  be  made  capable  of  resisting  con- 
siderable pressure.  As  an  example  the  New  Croton  aqueduct 
from  Croton  Lake  to  Jerome  Park  reservoir,  which  is  33*1  miles 


Fig.  121. 


Fig.  122. 


Fig.  123. 


Fig.  124. 


Fig.  125. 


Metal 


Fig.  127. 


Fig.  128. 


Fig.  129. 


Fig.  130. 


Fig.  131. 


218  HYDRAULICS 

long,  is  made  up  of  two  parts.  The  first  is  a  masonry  conduit  of 
the  section  shown  in  Fig.  121,  23'9  miles  long  and  having  a  slope 
of  '0001326,  the  second  consists  almost  entirely  of  a  brick  lined 
siphon  6'83  miles  long,  12'  3"  diameter,  the  maximum  head  in 
which  is  126  feet,  and  the  difference  in  level  of  the  two  ends  is 
6'19  feet.  In  such  cases,  however,  the  siphon  is  frequently  made 
of  steel,  or  cast-iron  pipes,  as  in  the  case  of  the  new  Edinburgh 
aqueduct  (see  Fig.  131)  which,  where  it  crosses  the  valleys,  is 
made  of  cast-iron  pipes  33  inches  diameter. 

135.     The  best  form  of  channel. 

The  best  form  of  channel,  or  channel  of  least  resistance,  is 
that  which,  for  a  given  slope  and  area,  will  give  the  maximum 
discharge. 

Since  the  mean  velocity  in  a  channel  of  given  slope  is  proper- 

j^ 
tional  to  p  ,  and  the  discharge  is  A  .  v,  the  best  form  of  channel  for 

a  given  area,  is  that  for  which  P  is  a  minimum. 

The  form  of  the  channel  which  has  the  minimum  wetted  peri- 
meter for  a  given  area  is  a  semicircle,  for  which,  if  r  is  the  radius> 

the  hydraulic  mean  depth  is  ~. 

More  convenient  forms,  for  channels  to  be  excavated  in  rock 
or  earth,  are  those  of  the  rectangular  or  trapezoidal  section> 
Fig.  133.  For  a  given  discharge,  the  best  forms  for  these 
channels,  will  be  those  for  which  both  A  and  P  are  a  minimum  ; 
that  is,  when  the  differentials  d  A  and  dP  are  respectively  equal  to 
zero. 

Rectangular  channel.  Let  L  be  the  width  and  h  the  depth, 
Fig.  132,  of  a  rectangular  channel  ;  it  is  required  to  find  the  ratio 

j-  that  the  area  A  and  the  wetted  perimeter  P  may  both  be  a 
minimum,  for  a  given  discharge. 
A  =  Lfc, 
therefore  3A  =  h.o~L+  ~Ldh  =  0    .....................  (1), 


therefore  3P  =  dL  +  2;)7i  =  0  ...........................  (2). 

Substituting  the  value  of  c*L  from  (2)  in  (1), 

L  =  27i. 

Therefore  m==l^=|- 

Since  L  =  27t,  the  sides  and  bottom  of  the  channel  touch  a  circle 
having  h  as  radius  and  the  centre  of  which  is  in  the  free  surface 
of  the  water. 


FLOW   IN   OPEN    CHANNELS 


219 


Earth  channels  of  trapezoidal  form.     In  Fig.  133  let 

I  be  the  bottom  width, 

h  the  depth, 

A  the  cross  sectional  area  FBCD, 

P  the  length  of  FBCD  or  the  wetted  perimeter, 

i  the  slope, 

and  let  the  slopes  of  the  sides  be  t  horizontal  to  one  vertical ;  CG 
is  then  equal  to  th  and  tan  CDGr  =  t. 


IT*  fiC 

Klt"  th—\ 


k-i-4 


Fig.  132. 


Fig.  133. 

Let  Q  be  the  discharge  in  cubic  feet  per  second. 

Then  A=hl  +  t~h? (3), 

~1  .  ...(4), 


and 


" 


.(5). 


For  the  channel  to  be  of  the  best  form  dP  and  dA  both  equal 
zero. 

From  (3)                 A  =  hl+  th\ 
and  therefore  dA  =  hdl  +  ldh  +  2thdh  =  0 (6). 

From  (4)  P  =  I 


and 


Substituting  the  value  of  dl  from  (7)  in  (6) 
Therefore,  n 


(7). 
.(8). 


1-2M 


h 
2* 


Let  O  be  the  centre  of  the  water  surface  AD,  then  since  from  (8) 


therefore,  in  Fig.  133,        CD  =  EG  -  OD. 


220  HYDRAULICS 

Draw  OF  and  OE  perpendicular  to  CD  and  BC  respectively. 

Then,  because  the  angle  OFD  is  a  right  angle,  the  angles  CDGr 
and  FOD  are  equal  ;  and  since  OF  -  OD  cos  FOD,  and  DG-  -  OE, 
and  Da  =  CDcosCDG,  therefore,  OE-OF;  and  since  OEC  and 
OFC  are  right  angles,  a  circle  with  0  as  centre  will  touch  the  sides 
of  the  channel,  as  in  the  case  of  the  rectangular  channel. 

136.  Depth  of  flow  in  a  channel  of  given  form  that, 
(a)  the  velocity  may  be  a  maximum,  (b)  the  discharge  may 
be  a  maximum. 

Taking  the  general  formula 

.     k.vn 


and  transposing, 


1    p 
inmn 


For  a  given  slope  and  roughness  of  the  channel  v  is,  therefore, 
proportional  to  the  hydraulic  mean  depth  and  will  be  a  maximum 
when  m  is  a  maximum. 

That  is,  when  the  differential  of  p  is  zero,  or 

For  maximum  discharge,  Aw  is  a  maximum,  and  therefore, 

P 
A\*. 


A     f  A 

MP) 


pi   is  a  maximum. 
Differentiating  and  equating  to  zero, 

—  PdA.  —  —  A.dP  =  0...  ...(2). 

n  n 

Affixing  values  to  n  and  p  this  differential  equation  can  be 
solved  for  special  cases.  It  will  generally  be  sufficiently  accurate 
to  assume  n  is  2  and  p  -  1,  as  in  the  Chezy  formula,  then 

n  +  p  _  3 
and  the  equation  becomes 

137.  Depth  of  flow  in  a  circular  channel  of  given 
radius  and  slope,  when  the  velocity  is  a  maximum. 

Let  r  be  the  radius  of  the  channel,  and  2^  the  angle  subtended 
by  the  surface  of  the  water  at  the  centre  of  the  channel,  Fig.  134. 


FLOW   IN   OPEN    CHANNELS  221 

Then  the  wetted  perimeter 


and 

/        sm  2(£\ 
The  area      A  =  r2</>  -  r2  sin  <f>  cos  <£  =  r2  {$ ^—  y  > 

and  dA  =  r*d<}>-r2  co$2<f>d<}>. 

Substituting   these   values   of   dP   and   dA  in    equation    (3), 
section  136, 


The  solution  in  this  case  is  obtained 
directly  as  follows, 

sin  i 


m  =  ^FT—  T^ 


This  will  be  a  maximum  when  sin 
is  negative,  and 


is  a  maximum,  or  when  Fig-  134- 

d   /sinf  ' 


and  tan  2</>  -  2<£. 

The  solution  to  this  equation,  for  which  2<£  is  less  than  360°,  is 

2<£  =  257°27'. 
Then  A  =  2'738r2, 


m  =  '608r, 
and  the  depth  of  flow  d  =  l'626r. 

138.  Depth  of  flow  in  a  circular  channel  for  maximum 
discharge. 

Substituting  for  dP  and  dA  in  equation  (3),  section  136, 

6r3<f>d<f>  -  er3^  cos  2<i>d<}>  -  2r3<W  +  r3  sin  2<W  -  0, 
from  which  4<£  -  6<£  cos  2^>  +  sin  2</>  =  0, 

and  therefore  <t>  =  154°. 

Then  A  =  3'044r2, 

P  =  5'30r, 
m  =  '573r, 
and  the  depth  of  flow  d  =  l'899r. 

Similar  solutions  can  be  obtained  for  other  forms  of  channels, 
and  may  be  taken  by  the  student  as  useful  mathematical  exercises 
but  they  are  not  of  much  practical  utility. 


222 


HYDRAULICS 


139.  Curves  of  velocity  and  discharge  for  a  given 
channel. 

The  depth  of  flow  for  maximum  velocity,  or  discharge,  can  be 
determined  very  readily  by  drawing  curves  of  velocity  and  dis- 
charge for  different  depths  of  flow  in  the  channel.  This  method 
is  useful  and  instructive,  especially  to  those  students  who  are  not 
familiar  with  the  differential  calculus. 

As  an  example,  velocities  and  discharges,  for  different  depths 
of  flow,  have  been  calculated  for  a  large  aqueduct,  the  profile  of 
which  is  shown  in  Fig.  135,  and  the  slope  i  of  which  is  0*0001326. 
The  velocities  and  discharges  are  shown  by  the  curves  drawn  in 
the  figure. 


Fig.  135. 

Values  of  A  and  P  for  different  depths  of  flow  were  first  deter- 
mined and  m  calculated  from  them. 

The  velocities  were  calculated  by  the  formula 

v  =  C  >Jmi, 

using  values  of  C  from  column  3,  Table  XXI. 

It  will  be  seen  that  the  velocity  does  not  vary  very  much  for 
all  depths  of  flow  greater  than  3  feet,  and  that  neither  the  velocity 
nor  the  discharge  is  a  maximum  when  the  aqueduct  is  full;  the 
reason  being  that,  as  in  the  circular  channel,  as  the  surface  of  the 
water  approaches  the  top  of  the  aqueduct  the  wetted  perimeter 
increases  much  more  rapidly  than  the  area. 

The  maximum  velocity  is  obtained  when  m  is  a  maximum 
and  equal  to  3'87,  but  the  maximum  discharge  is  given,  when  the 
depth  of  flow  is  greater  than  that  which  gives  the  greatest 


FLOW   IN    OPEN    CHANNELS  223 

Telocity.     A  circle  is  shown  on  the  figure  which  gives  the  same 
maximum  discharge. 

The  student  should  draw  similar  curves  for  the  egg-shaped 
sewer  or  other  form  of  channel. 

140.    Applications  of  the  formula. 

Problem  1.     To  find  the  flow  in  a  channel  of  given  section  and  slope. 

This  is  the  simplest  problem  and  can  be  solved  by  the  application  of  either  the 
logarithmic  formula  or  by  Bazin's  formula. 

The  only  difficulty  that  presents  itself,  is  to  affix  values  to  k,  n,  and  p  in  the 
logarithmic  formula  or  to  y  in  Bazin's  formula. 

(1)     By  the  logarithmic  formula. 

First  assign  some  value  to  k,  n,  and  p  by  comparing  the  lining  of  the  channel 
with  those  given  in  Tables  XXIV  to  XXXIII.  Let  w  be  the  cross  sectional  area  of 
the  water. 


.  v 


Then  since  i  = 

m'' 

log  v  =  -  log  i  +  — log  m  —  log  k, 

and  Q  —  u.v, 

or  log  Q  =  log  o>  +  -  log  i  +  -  log  m log  k. 

(2)     By  the  C he zy  formula,  using  Bazin's  coefficient. 

The  coefficient  for  a  given  value  of  m  must  be  first  calculated  from  the  formula 


or  taken  from  Table  XXI. 
Then 

and 

Example.     Determine  the  flow  in  a  circular  culvert  9  ft.  diameter,  lined  with 
smooth  brick,  the  slope  being  1  in  2000,  and  the  channel  half  full. 

Area  =  !?=2-25'. 


T.r  ^   , 

Wetted  perimeter     4 

(1)     By  the  logarithmic  formula 

ml*88 

i  =  -00073  -^T.. 
m115 

Therefore,   log  V=-L  log  -0005  +  ^  log  2 -25  -  -L  iog  -00007, 
v  =  4'55  ft.  per  sec., 
w  =  Tj!  =  31-8  sq.ft., 


Q  =  145  cubic  feet  per  sec. 

(2)     By  the  Chezy  formula,  using  Bazin's  coefficient, 
157  \5 

x/2-25 

'  =  132  Jfr25~^fa  =  4-43  ft.  per  sec. 
=  31-8  x  3-35  =  141  cubic  ft.  per  sec. 


224  HYDRAULICS 

Problem  2.  To  find  the  diameter  of  a  circular  channel  of  given  slope,  for  which 
the  maximum  discharge  is  Q  cubic  feet  per  second. 

The  hydraulic  mean  depth  m  for  maximum  discharge  is  -573r  '(section  138)  and 
A  =  3-044r2. 

Then  the  velocity  is  v  =  -757C  JrT, 

and  Q  =  2-370*^ 

and  the  diameter 

The  coefficient  C  is  unknown,  but  by  assuming  a  value  for  it,  an  approximation 
to  D  can  be  obtained  ;  a  new  value  for  C  can  then  be  taken  and  a  nearer  approxi- 
mation to  D  determined  ;  a  third  value  for  C  will  give  a  still  nearer  approximation 
to  D. 

Example.  A  circular  aqueduct  lined  with  concrete  has  a  diameter  of  5'  9"  and 
a  slope  of  1  foot  per  mile. 

To  find  the  diameter  of  two  cast-iron  siphon  pipes  5  miles  long,  to  be  put  in 
series  with  the  aqueduct,  and  which  shall  have  the  same  discharge  ;  the  difference 
of  level  between  the  two  ends  of  the  siphon  being  12-5  feet. 

The  value  of  m  for  the  brick  lined  aqueduct  of  circular  -section  when  the 
discharge  is  a  maximum  is  '573?'  =-64  feet. 

The  area  A-3'044r2  =  25  sq.  ft. 

Taking  C  as  130  from  Table  XXI  for  the  brick  culvert  and  110  for  the  cast-iron 
pipe  from  Table  XII,  then 


Therefore 


=  4-00  feet. 


Problem  3.  Having  given  the  bottom  width  I,  the  slope  i,  and  the  side  slopes  t 
of  a  trapezoidal  earth  channel,  to  calculate  the  discharge  for  a  given  depth. 

First  calculate  m  from  equation  (5),  section  135. 

From  Table  XXI  determine  the  corresponding  value  of  C,  or  calculate  C  from 
Bazin's  formula, 


then  v  =  C  >/mi, 

and  Q  =  A.v. 

A  convenient  formula  to  remember  is  the  approximate  formula  for  ordinary 
earth  channels 


For  values  of  m  greater  than  2,  v  as  calculated  from  this  formula  is  very  nearly 
equal  to  v  obtained  by  using  Bazin's  formula. 


The  formula 

m1'5 
may  also  be  used. 


FLOW   IN   OPEN   CHANNELS  225 

Example.     An  ordinary  earth  channel  has  a  width  1=  10  feet,  a  depth,  d  =  ±  feet, 
and  a  slope  i  =  -s-^-^.     Side  slopes  1  to  1.     To  find  Q 

A  =  46  sq.  ft., 
P  =  21-212ft., 
m  =  2-16  ft., 

•35-60-5' 
N/2716 
.•.  v  =  1-625  ft.  per  sec., 

Q  =  74-7  cubic  ft.  per  sec. 
From  the  formula 


v  =  1-63  ft.  per  sec., 
Q  =  75  cubic  ft.  per  sec., 
From  the  logarithmic  formula 

i  =  ^j^> 

v  =  1-649  ft.  per  sec., 

Q  =  75-8  cubic  feet  per  sec. 

Problem  4.     Having  given  the  flow  in  a  canal,  the  slope,  and  the  side  slopes,  to 
find  the  dimensions  of  the  profile  and  the  mean  velocity  of  flow, 

(a)  When  the  canal  is  of  the  best  form. 

(b)  When  the  depth  is  given. 

In  the  first  case  m  =  -  ,  and  from  equations  (8)  and  (4)  respectively,  section  135- 
2 


Therefore 
Substituting  -  for  m 


and  A2 

But  v  =  ?  =  CJmi. 

Therefore  C2  - 


A  value  for  C  should  be  chosen,  say  C  =  70,  and  h  calculated,  from  which  a  mean 
value  for  m  =  ^  can  be  obtained. 

A  nearer  approximation  to  h  can  then  be  determined  by  choosing  a  new  value  of  C, 
from  Table  XXI  corresponding  to  this  approximate  value  of  m,  and  recalculating 
h  from  equation  (1). 

Example.  An  earthen  channel  to  be  kept  in  very  good  condition,  having  a  slope 
of  1  in  10,000,  and  side  slopes  2  to  1,  is  required  to  discharge  100  cubic  feet 
per  second  ;  to  find  the  dimensions  of  the  channel ;  take  C  =  70. 

L.   H.  15 


226  HYDRAULICS 


Then  M         20'00° 


^900 
10,000 (      ' 
20,000 


•49  x  6-1 
=  6700, 

and  h= 5-4  feet. 

Therefore  m  =  2  -7. 

From  Table  XXI,  C  =  82  for  this  value  of  m,  therefore  a  nearer  approximation 
to  h  is  now  found  from 

M_     20,000      _  20,000 

82'  -  -67  x  6-1 ' 

x6'l 


10,000 
from  which  /t  =  5'22  ft.  and  m  =  2'61. 

The  approximation  is  now  sufficiently  near  for  all  practical  purposes  and  may 
be  taken  as  5£  feet. 

Problem  5.     Having  given  the  depth  d  of  a  trapezoidal  channel,  the  slope  i,  and 
the  side  slopes  t,  to  find  the  bottom  width  I  for  a  given  discharge. 
First  using  the  Chezy  formula, 


and 

The  mean  velocity 

Therefore  .  „  .  . 

In  this  equation  the  coefficient  C  is  unknown,  since  it  depends  upon  the  value 
of  m  which  is  unknown,  and  even  if  a  value  for  C  be  assumed  the  equation  cannot 
very  readily  be  solved.  It  is  desirable,  therefore,  to  solve  by  approximation. 

*  Assume  any  value  for  m,  and  find  from  column  4,  Table  XXI,  the  corresponding 
value  for  C,  and  use  these  values  of  m  and  C. 

Then,  calculate  v  from  the  formula 


Since  T 

A 

and  k 


Therefore  dl  +  td*  =      .,  ...(1). 

v 

From  this  equation  a  value  of  I  can  be  obtained,  which  will  probably  not  be  the 
correct  value. 

With  this  value  of  I  calculate  a  new  value  for  m,  from  the  formula 


For  this  value  of  m  obtain  a  new  value  of  C  from  the  table,  recalculate  v,  and 
by  substitution  in  formula  (1)  obtain  a  second  value  for  I. 

On  now  again  calculating  m  by  substituting  for  d  in  formula  (2),  it  will  generally 
be  found  that  m  differs  but  little  from  m  previously  calculated  ;  if  so,  the  approxi- 
mation has  proceeded  sufficiently  far,  and  d  as  determined  by  using  this  value  of  m 
will  agree  with  the  correct  value  sufficiently  nearly  for  all  practical  purposes. 

The  problem  can  be  solved  in  a  similar  way  by  the  logarithmic  formula 


The  indices  n  andp  may  be  taken  as  2-1,  and  1'5  respectively,  and  k  as  -00037. 


FLOW  IN   OPEN   CHANNELS  227 

Example.     The  depth  of  an  ordinary  earth  channel  is  4  feet,  the  side  slopes 
1  to  1,  the  slope  1  in  6000  and  the  discharge  is  to  be  7000  cubic  feet  per  minute. 
Find  the  bottom  width  of  the  channel. 
Assume  a  value  for  ra,  say  2  feet. 
From  the  logarithmic  formula 

2  -1  log  v  =  log  i  +  1-5  log  m-  4-5682  ...........................  (3), 

v  =  1-122  feet  per  sec. 
7000 

TheQ  A=dk6o=104s<»-feet- 

But  A 


Substituting  this  value  for  I  in  equation  (2) 


Kecalculating  v  from  formula  (3) 

Then  A  =  75*fee£ 

1=  14-75  feet, 
and  m  =  2-88  feet. 

The  first  value  of  I  is,  therefore,  too  large,  and  this  second  value  is  too  small. 
Third  values  were  found  to  be      v  =  1'455, 

A  =  80-2, 
J  =  16-05, 
m= 2-935. 
This  value  of  I  is  again  too  large. 

A  fourth  calculation  gave  v  =  1-475, 

A =79-2, 

m  =  2-92. 

The  approximation  has  been  carried  sufficiently  far,  and  even  further  than  is 
necessary,  as  for  such  channels  the  coefficient  of  roughness  k  cannot  be  trusted  to 
an  accuracy  corresponding  to  the  small  difference  between  the  third  and  fourth 
values  of  I. 

Problem  6.  Having  given  the  bottom  width  I,  the  slope  i  and  the  side  slopes  of 
a  trapezoidal  channel,  to  find  the  depth  d  for  a  given  discharge. 

This  problem  is  solved  exactly  as  the  last,  by  first  assumingji_value  for  m,  and 
calculating  an  approximate  value  for  v  from  the  formula  v  =  C*Jmi. 

Then,  by  substitution  in  equation  (1)  of  the  last  problem  and  solving  the 
quadratic, 


by  substituting  this  value  for  d  in  equation  (2),  a  new  value  for  m  can  be  found, 
and  hence,  a  second  approximation  to  d,  and  so  on. 

Using  the  logarithmic  formula  the  procedure  is  exactly  the  same  as  for 
problem  5. 

Problem  7  *.  Having  a  natural  stream  BC,  Fig.  135 a,  of  given  slope,  it  is  required 
to  determine  the  point  C,  at  which  a  canal,  of  trapezoidal  section,  which  is  to 
deliver  a  definite  quantity  of  water  to  a  given  point  A  at  a  given  level,  shall  be 
made  to  join  the  stream  so  that  the  cost  of  the  canal  is  a  minimum. 

*  The  solution  here  given  is  practically  the  same  as  that  given  by  M.  Flamant 
in  his  excellent  treatise  Hydraulique. 

15—2 


228  HYDRAULICS 

Let  I  be  the  slope  of  the  stream,  i  of  the  canal,  h  the  height  above  some  datum 
of  the  surface  of  the  water  at  A,  and  J^  of  the 
water  in  the  stream  at  B,  at  some  distance  L 
from  C. 

Let   L    be    also    the    length    and   A   the  |T>  \r* 

sectional   area  of  the  canal,   and  let  it   be  j  J\  ^ 

assumed  that  the  section  of  the  canal  is  of  the       A  j  y 

most  economical  form,  or  m=-. 

2  Fig.  135  a. 

The  side  slopes  of  the  canal  will  be  fixed 

according  to  the  nature  of  the  strata  through  which  the  canal  is  cut,  and  may  be 
supposed  to  be  known. 

Then  the  level  of  the  water  at  C  is 


Therefore  L  =  TZ- 

Let  I  be  the  bottom  width  of  the  canal,  and  t  the  slope  of  the  sides.     The  cross 
section  is  then  dl  +  td?,  and 

_A_       dl  +  tcP 

~P~ 

Substituting  2m  for  d, 

Z=4wN/«2+l-4tra, 

and  therefore  m= ==  =- , 

4m  tjf  + 1  -  4tm  +  4w  Jf2  + 1 


from  which  m2=- 

8 

The  coefficient  C  in  the  formula  v  =  G,Jmi  may  be  assumed  constant. 
Then  v2  =  C2wi, 

and  v4=C4w2i2. 

For  v  substituting  ^ ,  and  for  ra2  the  above  value, 
O4  C4Ai2 


and  A.H2=        (2  Jtf  +  l  -  1). 


Therefore  A  = 


The  cost  of  the  canal  will  be  approximately  proportional  to  the  product  of  the 
length  L  and  the  cross  sectional  area,  or  to  the  cubical  content  of  the  excavation. 
Let  £k  be  the  price  per  cubic  yard  including  buying  of  land,  excavation  etc.  Let  £x 
be  the  total  cost. 

Then  £c  =  £fc.L.A 

=  k.(h-hl) 

This  will  be  a  minimum  when  -TT  =  O. 

di 

Differentiating  therefore,  and  equating  to  zero, 

and  i  =  fl. 

The  most  economical  slope  is  therefore  f  of  the  slope  of  the  natural  stream. 

If  instead  of  taking  the  channel  of  the  best  form  the  depth  is  fixed,  the 
slope  i=£.I. 


FLOW  IN   OPEN   CHANNELS  229 

There  have  been  two  assumptions  made  in  the  calculation,  neither  of  which  is 
rigidly  true,  the  first  being  that  the  coefficient  C  is  constant,  and  the  second  that 
the  price  of  the  canal  is  proportional  to  its  cross  sectional  area. 

It  will  not  always  be  possible  to  adopt  the  slope  thus  found,  as  the  mean 
velocity  must  be  maintained  within  the  limits  given  on  page  216,  and  it  is  not 
advisable  that  the  slope  should  be  less  than  1  in  10,000. 


EXAMPLES. 

(1)  The  area  of  flow  in  a  sewer  was  found  to  be  0'28  sq.  feet;  the 
wetted  perimeter  T60  feet  ;  the  inclination  1  in  38*7.  The  mean  velocity 
of  flow  was  6*12  feet  per  second.  Find  the  value  of  C  in  the  formula 


(2)  The  drainage  area  of  a  certain  district  was  19*32  acres,  the  whole 
area  being  impermeable  to  rain  water.     The  maximum  intensity  of  the 
rainfall  was  0'360  ins.  per  hour  and  the  maximum  rate  of  discharge  regis- 
tered in  the  sewer  was  96%  of  the  total  rainfall. 

Find  the  size  of  a  circular  glazed  earthenware  culvert  having  a  slope  of 

1  in  50  suitable  for  carrying  the  storm  water. 

(3)  Draw  a  curve  of  mean  velocities  and  a  curve  of  discharge  for  an 
egg-shaped  brick  sewer,  using  Bazin's  coefficient.     Sewer,  6  feet  high  by 
4  feet  greatest  width  ;  slope  1  in  1200. 

(4)  The  sewer  of  the  previous  question  is  required  to  join  into  a  main 
outfall  sewer.     To  cheapen  the  junction  with  the  main  outfall  it  is  thought 
advisable  to  make  the  last  100  feet  of  the  sewer  of  a  circular  steel  pipe 
3  feet  diameter,  the  junction  between  the  oval  sewer  and  the  pipe  being 
carefully  shaped  so  that  there  is  no  impediment  to  the  flow. 

Find  what  fall  the  circular  pipe  should  have  so  that  its  maximum 
discharge  shall  be  equal  to  the  maximum  discharge  of  the  sewer.  Having 
found  the  slope,  draw  out  a  curve  of  velocity  and  discharge. 

(5)  A  canal  in  earth  has  a  slope  of  1  foot  in  20,000,  side  slopes  of 

2  horizontal  to   1  vertical,  a  depth   of  22  feet,  and  a  bottom  width  of 
200  feet;   find  the  volume  of  discharge. 

Bazin's  coefficient  -y=2'35. 

(6)  Give  the  diameter  of  a  circular  brick  sewer  to  run  half  -full  for  a 
population  of  80,000,  the  diurnal  volume  of  sewage  being  75  gallons  per 
head,  the  period  of  maximum  flow  6  hours,  and  the  available  fall  1  in  1000. 

Inst.  C.  E.  1906. 

(7)  A  channel  is  to  be  cut  with  side  slopes  of  1|  to  1  ;  depth  of  water, 

3  feet;  slope,  9  inches  per  mile:  discharge,  6,000  cubic  feet  per  minute. 
Find  by  approximation  dimensions  of  channel. 

(8)  An  area  of  irrigated  land  requires  2  cubic  yards  of  water  per  hour 
per  acre.    Find  dimensions  of  a  channel  3  feet  deep  and  with  a  side  slope 
of  1  to  1.     Fall,  1£  feet  per  mile.     Area  to  be  irrigated,  6000  acres.     (Solve 
by  approximation.)     y=2'35. 

(9)  A  trapezoidal  channel  in  earth  of  the  most  economical  form  has  a 
depth  of  10  feet  and  side  slopes  of  1  to  1.    Find  the  discharge  when  the 
slope  is  18  inches  per  mile.    y=2'35. 


230  HYDRAULICS 

(10)  A  river  has  the  following  section :— top  width,  800  feet ;  depth  of 
water,  20  feet ;  side  slopes  1  to  1 ;  fall,  1  foot  per  mile.    Find  the  discharge, 
using  Bazin's  coefficient  for  earth  channels. 

(11)  A  channel  is  to  be  constructed  for  a  discharge  of  2000  cubic  feet 
per  second ;  the  fall  is  1^  feet  per  mile ;  side  slopes,  1  to  1 ;  bottom  width, 
10  times  the  depth.     Find  dimensions  of  channel.     Use  the  approximate 


formula,  v- 

(12)  Find  the  dimensions  of  a  trapezoidal  earth  channel,  of  the  most 
economical  form,  to  convey  800  cubic  feet  per  second,  with  a  fall  of  2  feet 
per  mile,  and  side  slopes,  1^  to  1.     (Approximate  formula.) 

(13)  An  irrigation  channel,  with  side  slopes  of  l£  to  1,  receives  600 
cubic  feet  per  second.     Design  a  suitable  channel  of  3  feet  depth  and 
determine  its  dimensions  and  slope.     The  mean  velocity  is  not  to  exceed 
2£  feet  per  second.    y  =  2'35. 

(14)  A  canal,  excavated  in  rock,  has  vertical  sides,  a  bottom  width  of 
160  feet,  a  depth  of  22  feet,  and  the  slope  is  1  foot  in  20,000  feet.     Find  the 
discharge,    y  =  1  '54. 

(15)  A  length  of  the  canal  referred  to  in  question  (14)  is  in  earth.     It 
has  side  slopes  of  2  horizontal  to  1  vertical;  its  width  at  the  water  line 
is  290  feet  and  its  depth  22  feet. 

Find  the  slope  this  portion  of  the  canal  should  have,  taking  y  as  2-35. 

(16)  An  aqueduct  95|  miles  long  is  made  up  of  a  culvert  50 1  miles 
long  and  two  steel  pipes  3  feet  diameter  and  45  miles  long  laid  side  by  side. 
The  gradient  of  the  culvert  is  20  inches  to  the  mile,  and  of  the  pipes  2  feet 
to  the  mile.    Find  the  dimensions  of  a  rectangular  culvert  lined  with  well 
pointed  brick,  so  that  the  depth  of  flow  shall  be  equal  to  the  width  of  the 
culvert,  when  the  pipes  are  giving  their  maximum  discharge. 

Take  for  the  culvert  the  formula 

•000061  v1'88 


1-15 


m 
and  for  the  pipes  the  formula 

._  -00050.  v2 
d1'26      ' 

(17)  The  Ganges  canal  at  Taoli  was  found  to  have  a  slope  of  0*000146 
and  its  hydraulic  mean  depth  m  was  7'0  feet ;  the  velocity  as  determined 
by  vertical  floats  was  2-80  feet  per  second;  find  the  value  of  C  and  the 
value  of  y  in  Bazin's  equation. 

(18)  The  following  data  were  obtained  from  an  aqueduct  lined  with 
brick  carefully  pointed : 

m                                  i  v 

in  metres  in  metres  per  sec. 

•229                        0*0001326  '336 

•381                               „  -484 

•533                               „  -596 

•686                                „  -691 

•838                                „  -769 

•991                                „  -848 

1-143                                „  -913 

1-170  -922 


FLOW  IN   OPEN   CHANNELS  231 


Plot  -=  as  ordinates,  --  as  abscissae  ;  find  values  of  a  and  8  in  Bazin's 

Vra  v 

formula,  and  thus  deduce  a  value  of  y  for  this  aqueduct. 

(19)  An  aqueduct  107^  miles  long  consists  of  13J  miles  of  siphon,  and 
the  remainder  of  a  masonry  culvert  6  feet  10^  inches  diameter  with  a  gradient 
of  1  in  8000.     The  siphons  consist  of  two  lines  of  cast-iron  pipes  43  inches 
diameter  having  a  slope  of  1  in  500.     Determine  the  maximum  discharge. 

(20)  An  aqueduct  consists  partly  of  the  section  shown  in  Fig.  131, 
page  217,  and  partly  (i.e.  when  crossing  valleys)  of  33  inches  diameter  cast- 
iron  pipe  siphons. 

Determine  the  minimum  slope  of  the  siphons,  so  that  the  aqueduct 
may  discharge  15,000,000  gallons  per  day,  and  the  slope  of  the  masonry 
aqueduct  so  that  the  water  shall  not  be  more  than  4  feet  6  inches  deep  in 
the  aqueduct. 

(21)  Calculate  the  quantity  delivered  by  the  water  main  in  question  (30)  , 
page  172,  per  day  of  24  hours. 

This  amount,  representing  the  water  supply  of  a  city,  is  discharged  into 
the  sewers  at  the  rate  of  one-half  the  total  daily  volume  in  6  hours,  and  is 
then  trebled  by  rainfall.  Find  the  diameter  of  the  circular  brick  outfall 
sewer  which  will  carry  off  the  combined  flow  when  running  half  full,  the 
available  fall  being  1  in  1500.  Use  Bazin's  coefficient  for  brick  channels. 

(22)  Determine    for    a   smooth  cylindrical  cast-iron  pipe  the  angle 
subtended  at  the  centre  by  the  wetted  perimeter,  when  the  velocity  of  flow 
is  a  maximum.     Determine  the  hydraulic  mean  depth  of  the  pipe  under 
these  conditions.     Lond.  Un.  1905. 

(23)  A  9  -inch  drain  pipe  is  laid  at  a  slope  of  1  in  150,  and  the  value  of 
c  is  107  (v  =  c>Jmi).     Find  a  general  expression  for  the  angle  subtended  at 
the  centre  by  the  water  line,  and  the  velocity  of  flow;  and  indicate  how  the 
general  equations  may  be  solved  when  the  discharge  is  given.     Lond.  Un. 
1906. 

141.  Short  account  of  the  historical  development  of  the  pipe  and  channel  formulae. 
It  seems  remarkable  that,  although  the  practice  of  conducting  water  along  pipes 
and  channels  for  domestic  and  other  purposes  has  been  carried  on  for  many 
centuries,  no  serious  attempt  to  discover  the  laws  regulating  the  flow  seems 
to  have  been  attempted  until  the  eighteenth  century.  It  seems  difficult  to  realise 
how  the  gigantic  schemes  of  water  distribution  of  the  ancient  cities  could  have  been 
executed  without  such  knowledge,  but  certain  it  is,  that  whatever  information  they 
possessed,  it  was  lost  during  the  middle  ages. 

It  is  of  peculiar  interest  to  note  the  trouble  taken  by  the  Roman  engineers  in 
the  construction  of  their  aqueducts.  In  order  to  keep  the  slope  constant  they 
tunnelled  through  hills  and  carried  their  aqueducts  on  magnificent  arches.  The 
Claudian  aqueduct  was  38  miles  long  and  had  a  constant  slope  of  five  feet  per  mile. 
Apparently  they  were  unaware  of  the  simple  fact  that  it  is  not  necessary  for  a  pipe 
or  aqueduct  connecting  two  reservoirs  to  be  laid  perfectly  straight,  or  else  they 
wished  the  water  at  all  parts  of  the  aqueducts  to  be  at  atmospheric  pressure. 

Stephen  Schwetzer  in  his  interesting  treatise  on  hydrostatics  and  hydraulics 
published  in  1729  quotes  experiments  by  Marriott  showing  that,  a  pipe  1400  yards 
long,  1|  inches  diameter,  only  gave  £  of  the  discharge  which  a  hole  If  inches  diameter 
in  the  side  of  a  tank  would  give  under  the  same  head,  and  also  explains  that  the 
motion  of  the  liquid  in  the  pipes  is  diminished  by  friction,  but  he  is  entirely 
ignorant  of  the  laws  regulating  the  flow  of  fluids  through  pipes.  Even  as  late  as 


232  HYDRAULICS 

1786  Du  Buat*  wrote,  "We  are  yet  in  absolute  ignorance  of  the  laws  to  which  the 
movement  of  water  is  subjected." 

The  earliest  recorded  experiments  of  any  value  on  long  pipes  are  those  of 
Couplet,  in  which  he  measured  the  flow  through  the  pipes  which  supplied  the 
famous  fountains  of  Versailles  in  1732.  In  1771  Abbe"  Bossut  made  experiments  on 
flow  in  pipes  and  channels,  these  being  followed  by  the  experiments  of  Du  Buat,  who 
erroneously  argued  that  the  loss  of  head  due  to  friction  in  a  pipe  was  independent 
of  the  internal  surface  of  the  pipe,  and  gave  a  complicated  formula  for  the  velocity 
of  flow  when  the  head  and  the  length  of  the  pipe  were  known. 

In  1775  M.  Chezy  from  experiments  upon  the  flow  in  an  open  canal,  came  to 
the  conclusion  that  the  fluid  friction  was  proportional  to  the  velocity  squared,  and 
that  the  slope  of  the  channel  multiplied  by  the  cross  sectional  area  of  the  stream, 
was  equal  to  the  product  of  the  length  of  the  wetted  surface  measured  on  the  cross 
section,  the  velocity  squared,  and  some  constant,  or 

iK  =  Pavz    (1), 

i  being  the  slope  of  the  bed  of  the  channel,  A  the  cross  sectional  area  of  the  stream, 
P  the  wetted  perimeter,  and  a  a  coefficient. 

From  this  is  deduced  the  well-known  Chezy  formula 


Pronyf,  applying  to  the  flow  of  water  in  pipes  the  results  of  the  classical  experi- 
ments of  Coulomb  on  fluid  friction,  from  which  Coulomb  had  deduced  the  law  that 
fluid  friction  was  proportional  to  av  +  bv2,  arrived  at  the  formula 


=  f- 


This  is  similar  to  the  Chezy  formula,  I  -  +  0  j  being  equal  to  ^ . 

By  an  examination  of  the  experiments  of  Couplet,  Bossut,  and  Du  Buat,  Prony 
gave  values  to  a  and  /3  which  when  transformed  into  British  units  are, 

a  —00001733, 
p=  -00010614. 

For  velocities,  above  2  feet  per  second,  Prony  neglected  the  term  containing  the 
first  power  of  the  velocity  and  deduced  the  formula 

v  =  48-6^/^77. 

He  continued  the  mistake  of  Du  Buat  and  assumed  that  the  friction  was  in- 
dependent of  the  condition  of  the  internal  surface  of  the  pipe  and  gave  the  following 
explanation :  "  When  the  fluid  flow?  in  a  pipe  or  upon  a  wetted  surface  a  film  of 
fluid  adheres  to  the  surface,  and  this  film  may  be  regarded  as  enclosing  the  mass 
of  fluid  in  motion  J."  That  such  a  film  encloses  the  moving  water  receives  support 
from  the  experiments  of  Professor  Hele  Shaw§.  The  experiments  were  made  upon 
such  a  small  scale  that  it  is  difficult  to  say  how  far  the  results  obtained  are  indica- 
tive of  the  conditions  of  flow  in  large  pipes,  and  if  the  film  exists  it  does  not  seem 
to  act  in  the  way  argued  by  Prony. 

TT 

The  value  of  i  in  Prony's  formula  was  equal  to  — ,  H  including,  not  only  the 

loss  of  head  due  to  friction  but,  as  measured  by  Couplet,  Bossut  and  Du  Buat, 
it  also  included  the  head  necessary  to  give  velocity  to  the  water  and  to  overcome 
resistances  at  the  entrance  to  the  pipe. 

Eytelwein  and  also  Aubisson,  both  made  allowances  for  these  losses,  by  sub- 
tracting from  H  a  quantity  — ,  and  then  determined  new  values  for  a  and  b  in  the 

&9 
formula 


*  Le  Discours  preliminaire  de  ses  Principes  d'hydr antique. 

t  See  also  Girard's  Movement  des  fluids  dans  les  tubes  capillaires,  1817. 

J  Traite  d'hydraulique.  §  Engineer,  Aug.  1897  and  May  1898. 


FLOW  IN   OPEN   CHANNELS  233 

They  gave  to  a  and  b  the  following  values. 

Eytelwein     a  =  -000023584, 

6  =-000085434. 

Aubisson*     a  =  -000018837, 

6  =-000104392. 

By  neglecting  the  term  containing  v  to  the  first  power,  and  transforming  the 
terms,  Aubisson's  formula  reduces  to 


I  +  35-5d  ' 

Young,  in  the  Encyclopaedia  Britannica,  gave  a  complicated  formula  for  v  when 
H  and  d  were  known,  but  gave  the  simplified  formula,  for  velocities  such  as 
are  generally  met  with  in  practice, 


St 


^ 

Venant  made  a  decided  departure  by  making  -  proportional  to  v~?~  instead  of 


to  v2  as  in  the  Chezy  formula. 

When  expressed  in  English  feet  as  units,  his  formula  becomes 


Weisbach  by  an  examination  of  the  early  experiments  together  with  ten  others  by 
himself  and  one  by  M.  Gueynard  gave  to  the  coefficient  a  in  the  formula  h  =  — 

771 

the  value 


that  is,  he  made  it  to  vary  with  the  velocity. 
Then,  mi 


the  values  of  a  and  ft  being  a  =  0-0144, 

j3  =  0-01716. 

From  this  formula  tables  were  drawn  up  by  Weisbach,  and  in  England  by 
Hawkesley,  which  were  considerably  used  for  calculations  relating  to  flow  of 
water  in  pipes. 

Darcy,  as  explained  in  Chapter  V,  made  the  coefficient  a  to  vary  with  the 
diameter,  and  Hagen  proposed  to  make  it  vary  with  both  the  velocity  and  the 
diameter. 

His  formula  then  became 

The  formulae  of  Ganguillet  and  Kutter  and  of  Bazin  have  been  given  in 
Chapters  V  and  VI. 

Dr  Lampe  from  experiments  on  the  Dantzig  mains  and  other  pipes  proposed 
the  formula 


thus  modifying  St  Venant's  formula  and  anticipating  the  formulae  of  Eeynolds, 
Flamant  and  Unwin,  in  which, 


n  and  p  being  variable  coefficients. 

*  Traite  d'hydraulique. 


CHAPTER  VII. 

GAUGING   THE   FLOW  OF  WATER. 

142.  Measuring  the  flow  of  water  by  weighing. 

In  the  laboratory  or  workshop  a  flow  of  water  can  generally 
be  measured  by  collecting  the  water  in  tanks,  and  either  by 
direct  weighing,  or  by  measuring  the  volume  from  the  known 
capacity  of  the  tank,  the  discharge  in  a  given  time  can  be 
determined.  This  is  the  most  accurate  method  of  measuring 
water  and  should  be  adopted  where  possible  in  experimental 
work. 

In  pump  trials  or  in  measuring  the  supply  of  water  to  boilers, 
determining  the  quantity  by  direct  weighing  has  the  distinct 
advantage  that  the  results  are  not  materially  affected  by 
changes  of  temperature.  It  is  generally  necessary  to  have  two 
tanks,  one  of  which  is  filling  while  the  other  is  being  weighed 
and  emptied.  For  facility  in  weighing  the  tanks  should  stand 
on  the  tables  of  weighing  machines. 

143.  Meters. 

Liner t  meter.  An  ingenious  direct  weighing  meter  suitable  for 
gauging  practically  any  kind  of  liquid,  is  constructed  as  shown  in 
Figs.  136  and  137. 

It  consists  of  two  tanks  A1  and  A2,  each  of  which  can  swing 
on  knife  edges  BB.  The  liquid  is  allowed  to  fall  into  a  shoot  F, 
which  swivels  about  the  centre  J,  and  from  which  it  falls  into 
either  A1  or  A2  according  to  the  position  of  the  shoot.  The  tanks 
have  weights  D  at  one  end,  which  are  so  adjusted  that  when  a 
certain  weight  of  water  has  run  into  a  tank,  it  swings  over  into 
the  dotted  position,  Fig.  136,  and  flow  commences  through  a 
siphon  pipe  C.  When  the  level  of  the  liquid  in  the  tank  has 
fallen  sufficiently,  the  weights  D  cause  the  tank  to  come  back  to 
its  original  position,  but  the  siphon  continues  in  action  until  the 
tank  is  empty.  As  the  tank  turns  into  the  dotted  position 


GAUGING   THE   FLOW  OF   WATER 


235 


it  suddenly  tilts  over  the  shoot  F,  and  the  liquid  is  discharged 
into  the  other  tank.  An  indicator  H  registers  the  number  of 
times  the  tanks  are  filled,  and  as  at  each  tippling  a  definite  weight 
of  fluid  is  emptied  from  the  tank,  the  indicator  can  be  marked 
off  in  pounds  or  in  any  other  unit. 


Fig.  136.  Fig.  137. 

Linert  direct  weighing  meter. 

144.    Measuring  the  flow  by  means  of  an  orifice. 

The  coefficient  of  discharge  of  sharp-edged  orifices  can  be 
obtained,  with  considerable  precision,  from  the  tables  of  Chapter  IY, 
or  the  coefficient  for  any  given  orifice  can  be  determined  for 
various  heads  by  direct  measurement  of  the  flow  in  a  given  time, 
as  described  above.  Then,  knowing  the  coefficient  of  discharge  at 
various  heads  a  curve  of  rate  of  discharge  for  the  orifice,  as  in 
Fig.  138,  may  be  drawn,  and  the  orifice  can  then  be  used  to 
measure  a  continuous  flow  of  water. 

The  orifice  should  be  made  in  the  side  or  bottom  of  a  tank.  If 
in  the  side  of  the  tank  the  lower  edge  should  be  at  least  one  and 
a  half  to  twice  its  depth  above  the  bottom  of  the  tank,  and  the 
sides  of  the  orifice  whether  horizontal  or  vertical  should  be  at 
least  one  and  a  half  to  twice  the  width  from  the  sides  of  the  tank. 
The  tank  should  be  provided  with  baffle  plates,  or  some  other 
arrangement,  for  destroying  the  velocity  of  the  incoming  water 
and  ensuring  quiet  water  in  the  neighbourhood  of  the  orifice.  The 
coefficient  of  discharge  is  otherwise  indefinite.  The  head  over  the 
orifice  should  be  observed  at  stated  intervals.  A  head-time  curve 
having  head  as  ordinates  and  time  as  abscissae  can  then  be  plotted 
as  in  Fig.  139. 

From  the  head-discharge  curve  of  Fig.  138  the  rate  of  discharge 
can  be  found  for  any  head  h,  and  the  curve  of  Fig.  139  plotted. 
The  area  of  this  curve  between  any  two  ordinates  AB  and  CD, 


236 


HYDRAULICS 


which  is  the  mean  ordinate  between  AB  and  CD  multiplied  by  the 
time  t,  gives  the  discharge  from  the  orifice  in  time  t. 

The  head  h  can  be  measured  by  fixing  a  scale,  having  its  zero 
coinciding  with  the  centre  of  the  orifice,  behind  a  tube  on  the  side 
of  the  tank. 


Fig.   138. 


B  TUne          D 

Fig.  139. 


B 

A 

E 

D 

^xf 

/                             x\ 

A 

Fig.  140. 


145.    Measuring  the  flow  in  open  channels. 

Large  open  channels  :  floats.  The  oldest  and  simplest  method 
of  determining  approximately  the  discharge  in  an  open  channel  is 
by  means  of  floats. 

A  part  of  the  channel  as  straight  as  possible  is  selected,  and  in 
which  the  flow  may  be  considered  as  uniform. 

The  readings  should  be  taken  on  a  calm  day  as  a  down-stream 
wind  will  accelerate  the  floats  and  an  up-stream  wind  retard  them. 

Two  cords  are  stretched  across  the  channel,  as  near  to  the 
surface  as  possible,  and  perpendicular  to  the  direction  of  flow.  The 
distance  apart  of  the  cords  should  be  as  great  as  possible  consistent 
with  uniform  flow,  and  should  not  be  less  than  150  feet.  From  a 
boat,  anchored  at  a  point  not  less  than  50  to  70  feet  above  stream, 
so  that  the  float  shall  acquire  before  reaching  the  first  line  a 
uniform  velocity,  the  float  is  allowed  to  fall  into  the  stream  and 


GAUGING  THE  FLOW  OF  WATER  237 

the  time  carefully  noted  by  means  of  a  chronometer  at  which  it 
passes  both  the  first  and  second  line.  If  the  velocity  is  slow,  the 
observer  may  walk  along  the  bank  while  the  float  is  moving  from 
one  cord  to  the  other,  but  if  it  is  greater  than  200  feet  per  minute 
two  observers  will  generally  be  required,  one  at  each  line. 

A  better  method,  and  one  which  enables  any  deviation  of  the 
float  from  a  path  perpendicular  to  the  lines  to  be  determined,  is, 
for  two  observers  provided  with  box  sextants,  or  theodolites,  to  be 
stationed  at  the  points  A  and  B,  which  are  in  the  planes  of  the 
two  lines.  As  the  float  passes  the  line  AA  at  D,  the  observer 
at  A  signals,  and  the  observer  at  B  measures  the  angle  ABD 
and,  if  both  are  provided  with  watches,  each  notes  the  time. 
When  the  float  passes  the  line  BB  at  E,  the  observer  at  B  signals, 
and  the  observer  at  A  measures  the  angle  BAE,  and  both 
observers  again  note  the  time.  The  distance  DE  can  then  be 
accurately  determined  by  calculation  or  by  a  scale  drawing,  and 
the  mean  velocity  of  the  float  obtained,  by  dividing  by  the  time. 

To  ensure  the  mean  velocities  of  the  floats  being  nearly  equal 
to  the  mean  velocity  of  the  particles  of  water  in  contact  with 
them,  their  horizontal  dimensions  should  be  as  small  as  possible, 
so  as  to  reduce  friction,  and  the  portion  of  the  float  above  the 
surface  of  the  water  should  be  very  small  to  diminish  the  effect  of 
the  wind. 

As  pointed  out  in  section  130,  the  distribution  of  velocity  in 
any  transverse  section  is  not  by  any  means  uniform  and  it  is 
necessary,  therefore,  to  obtain  the  mean  velocity  on  a  number  of 
vertical  planes,  by  finding  not  only  the  surface  velocity,  but  also 
the  velocity  at  various  depths  on  each  vertical. 

146.  Surface  floats. 

Surface  floats  may  consist  of  washers  of  cork,  or  wood,  or 
other  small  floating  bodies,  weighted  so  as  to  just  project  above 
the  water  surface.  The  surface  velocity  is,  however,  so  likely  to 
be  aifected  by  wind,  that  it  is  better  to  obtain  the  velocity  a 
short  distance  below  the  surface. 

147.  Double  floats. 

To  measure  the  velocity  at  points  below  the  surface  double 
floats  are  employed.  They  consist  of  two  bodies  connected  by 
means  of  a  fine  wire  or  cord,  the  upper  one  being  made  as  small 
as  possible  so  as  to  reduce  its  resistance. 

Gordon*,  on  the  Irrawaddi,  used  two  wooden  floats  connected 
by  a  fine  fishing  line,  the  lower  float  being  a  cylinder  1  foot  long, 

*  Proc.Inst.  C.  E.,  1893. 


238  HYDRAULICS 

and  6  inches  diameter,  hollow  underneath  and  loaded  with  clay  to 
sink  it  to  any  required  depth ;  the  upper  float,  which  swam  on  the 
surface,  was  of  light  wood  1  inch  thick,  and  carried  a  small  flag. 

The  surface  velocity  was  obtained  by  sinking  the  lower  float 
to  a  depth  of  3J  feet,  the  velocity  at  this  depth  being  not  very 
different  from  the  surface  velocity  and  the  motion  of  the  float  more 
independent  of  the  effect  of  the  wind. 


Fig.  141.     Gurley's  current  meter. 

Subsurface  velocities  were  measured  by  increasing  the  depths 
of  the  lower  float  by  lengths  of  3j  feet  until  the  bottom  was 
reached. 


GAUGING  THE  FLOW  OF  WATER  239 

Gordon  has  compared  the  results  obtained  by  floats  with  those 
obtained  by  means  of  a  current  meter  (see  section  149).  For 
small  depths  and  low  velocities  the  results  obtained  by  double 
floats  are  fairly  accurate,  but  at  high  velocities  and  great  depths, 
the  velocities  obtained  are  too  high.  The  error  is  from  0  to  10 
per  cent. 

Double  floats  are  sometimes  made  with  two  similar  floats,  of 
the  same  dimensions,  one  of  which  is  ballasted  so  as  to  float  at  any 
required  depth  and  the  other  floats  just  below  the  surface.  The 
velocity  of  the  float  is  then  the  mean  of  the  surface  velocity 
and  the  velocity  at  the  depth  of  the  lower  float. 

148.  Rod  floats. 

The  mean  velocity,  on  any  vertical,  may  be  obtained  ap- 
proximately by  means  of  a  rod  float,  which  consists  of  a  long  rod 
having  at  the  lower  end  a  small  hollow  cylinder,  which  may  be 
filled  with  lead  or  other  ballast  so  as  to  keep  the  rod  nearly 
vertical. 

The  rod  is  made  sufficiently  long,  and  the  ballast  adjusted,  so 
that  its  lower  end  is  near  to  the  bed  of  the  stream,  and  its  upper 
end  projects  slightly  above  the  water.  Its  velocity  is  approximately 
the  mean  velocity  in  the  vertical  plane  in  which  it  floats. 

149.  The  current  meter. 

The  discharge  of  large  channels  or  rivers  can  be  obtained  most 
conveniently  and  accurately  by  determining  the  velocity  of  flow 
at  a  number  of  points  in  a  transverse  section  by  means  of  a  current 
meter. 

The  arrangement  shown  in  Fig.  141  is  a  meter  of  the  anemo- 
meter type.  A  wheel  is  mounted  on  a  vertical  spindle  and  has 
five  conical  buckets.  The  spindle  revolves  in  bearings,  from 
which  all  water  is  excluded,  and  which  are  carefully  made  so 
that  the  friction  shall  remain  constant.  The  upper  end  of  the 
spindle  extends  above  its  bearing,  into  an  air-tight  chamber,  and 
is  shaped  to  form  an  eccentric.  A  light  spring  presses  against 
the  eccentric,  and  successively  makes  and  breaks  an  electric 
circuit  as  the  wheel  revolves.  The  number  of  revolutions  of  the 
wheel  is  recorded  by  an  electric  register,  which  can  be  arranged 
at  any  convenient  distance  from  the  wheel.  When  the  circuit  is 
made,  an  electro-magnet  in  the  register  moves  a  lever,  at  the  end 
of  which  is  a  pawl  carrying  forward  a  ratchet  wheel  one  tooth 
for  each  revolution  of  the  spindle.  The  frame  of  the  meter,  which 
is  made  of  bronze,  is  pivoted  to  a  hollow  cylinder  which  can  be 
clamped  in  any  desired  position  to  a  vertical  rod.  At  the  right- 


240  HYDRAULICS 

hand  side  is  a  rudder  having  four  light  metal  wings,  which 
balances  the  wheel  and  its  frame.  When  the  meter  is  being  used 
in  deep  waters  it  is  suspended  by  means  of  a  fine  cable,  and  to 
the  lower  end  of  the  rod  is  fixed  a  lead  weight.  The  electric 
circuit  wires  are  passed  through  the  trunnion  and  so  have  no 
tendency  to  pull  the  meter  out  of  the  line  of  current.  When 
placed  in  a  current  the  meter  is  free  to  move  about  the  horizontal 
axis,  and  also  about  a  vertical  axis,  so  that  it  adjusts  itself  to 
the  direction  of  the  current. 

The  meters  are  rated  by  experiment  and  the  makers  recommend 
the  following  method.  The  meter  should  be  attached  to  the  bow 
of  a  boat,  as  shown  in  Fig.  142,  and  immersed  in  still  water  not 
less  than  two  feet  deep.  A  thin  rope  should  be  attached  to  the 
boat,  and  passed  round  a  pulley  in  line  with  the  course  in  which 
the  boat  is  to  move.  Two  parallel  lines  about  200  feet  apart 
should  be  staked  on  shore  and  at  right  angles  to  the  course  of  the 
boat.  The  boat  should  be  without  a  rudder,  but  in  the  boat  with 
the  observer  should  be  a  boatman  to  keep  the  boat  from  running 


Fig.  142. 

into  the  shore.  The  boat  should  then  be  hauled  between  the  two 
ranging  lines  at  varying  speeds,  which  during  each  passage  should 
be  as  uniform  as  possible.  With  each  meter  a  reduction  table  is 
supplied  from  which  the  velocity  of  the  stream  in  feet  per  second 
can  be  at  once  determined  from  the  number  of  revolutions  recorded 
per  second  of  the  wheel. 

The  Haskell  meter  has  a  wheel  of  the  screw  propeller  type 
revolving  upon  a  horizontal  axis.  Its  mode  of  action  is  very 
similar  to  the  one  described. 

Comparative  tests  of  the  discharges  along  a  rectangular  canal 
as  measured  by  these  two  meters  and  by  a  sharp-edged  weir  which 
had  been  carefully  calibrated,  in  no  case  differed  by  more  than 
5  per  cent,  and  the  agreement  was  generally  much  closer*. 

*  Murphy  on  current  Meter  and  Weir  discharges,  Proceedings  Am.S.C.E., 
Vol.  xxvii.,  p.  779. 


GAUGING   THE   FLOW   OF   WATER 


241 


150.     Pitot  tube. 

Another  apparatus  which  can  be  used  for  determining  the 
velocity  at  a  point  in  a  flowing  stream,  even  when  the  stream  is  of 
small  dimensions,  as  for  example  a  small  pipe,  is  called  a  Pitot 
tube. 

In  its  simplest  form,  as  originally  proposed  by  Pitot  in  1732, 
it  consists  of  a  glass  tube,  with  a 
small  orifice  at  one  end  which  may 
be  turned  to  receive  the  impact  of 
the  stream  as  shown  in  Fig.  143. 
The  water  in  the  tube  rises  to  a 
height  h  above  the  free  surface  of 
the  water,  the  value  of  h  depending 
upon  the  velocity  v  at  the  orifice  of 
the  tube.  If  a  second  tube  is  placed 


h 


1^0 

Fig.  143.     Pitot  tube. 


beside  the  first  with  an  orifice  0  parallel  to  the  direction  of  flow, 
the  water  will  rise  in  this  tube  nearly  to  the  level  of  the  free 
surface,  the  fall  hi  being  due  to  a  slight  diminution  in  pressure 
at  the  mouth  of  the  tube,  caused  probably  by  the  stream  lines, 
having  their  directions  changed  at  the  mouth  of  the  tube.  A. 
further  depression  of  the  free  surface  in  the  tube  takes  place,, 
if  the  tube,  as  EF,  is  turned  so  that  the  orifice  faces  down  stream.. 

Theory  of  the  Pitot  tube.  Let  v  be  the  velocity  of  the  stream 
at  the  orifice  of  the  tube  in  ft.  per  sec.  and  a  the  area  of  the 
orifice  in  sq.  ft. 

The  quantity  of  water  striking  the  orifice  per  second  is  wav 
pounds. 


w 


The  momentum  is  therefore  -  .  a .  v*  pounds  feet. 

If  the  momentum  of  this  water  is  entirely  destroyed,  the 
pressure  on  the  orifice  which,  according  to  Newton's  second  law  of 
motion  is  equal  to  the  rate  of  change  of  momentum,  is 


-p  _ 


wav 


and  the  pressure  per  unit  area  is 


The  equivalent  head 


,  _  WIT  _  V' 
il/  —       •  —       . 
WO         Cf 


According  to  this  theory,  the  head  of  water  in  the  tube,  due  to 

v* 
the  impact,  is  therefore  twice  ~-  >  the  head  due  to  the  velocity  v,  and 


L.  H. 


16 


242 


HYDRAULICS 


the  water  should  rise  in  the  tube  to  a  height  above  the  surface 
equal  to  h. 

Experiment  shows  that  the  actual  height  the  water  rises  in  the 

v2  v2 

tube  is  more  nearly  equal  to  the  velocity  head  ^-  than   to   —  , 

•s  & 

and  the  head  h  is  thus  generally  taken  as 


c  being  a  coefficient  for  any  given  tube,  which  experiment  shows 
is  fairly  constant. 

Similarly  for  given  tubes 


and 


,        2 

2  =  ~ 


The  coefficients  are  determined  by  placing  the  tubes  in  streams 
the  velocities  of  which  are  known,  or  by  attaching  them  to  some 
body  which  moves  through  still  water  with  a  known  velocity,  and 
carefully  measuring  h  for  different  velocities. 


B 


D 


Fig.  144. 


Fig.  145. 


Darcy*  was  the  first  to  use  the  Pitot  tube  as  an  instrument  of 
precision.  His  improved  apparatus  as  ^used  in  open  channels  con- 
sisted of  two  tubes  placed  side  by  side  as  in  Fig.  144,  the  orifices 
in  the  tubes  facing  up-stream  and  down-stream  respectively.  The 


Eecherches  Hydrauliques,  etc.,  1857. 


GAUGING  THE   FLOW  OF   WATER  243 

two  tubes  were  connected  at  the  top,  a  cock  C1  being  placed  in  the 
common  tube  to  allow  the  tubes  to  be  opened  or  closed  to  the 
atmosphere.  At  the  lower  end  both  tubes  could  be  closed  at  the 
same  time  by  means  of  cock  C.  When  the  apparatus  is  put  into 
flowing  water,  the  cocks  C  and  C1  being  open,  the  free  surface 
rises  in  the  tube  B  a  height  hi  and  is  depressed  in  D  an  amount 
Tia.  The  cock  C1  is  then  closed,  and  the  apparatus  can  be  taken 
from  the  water  and  the  difference  in  the  level  of  the  two  columns, 

h  =  hi  +  h%, 
measured  with  considerable  accuracy. 

If  desired,  air  can  be  aspirated  from  the  tubes  and  the  columns 
made  to  rise  to  convenient  levels  for  observation,  without  moving 
the  apparatus.  The  difference  of  level  will  be  the  same,  whatever 
the  pressure  in  the  upper  part  of  the  tubes. 

Fig.  145  shows  one  of  the  forms  of  Pitot  tubes,  as  experimented 
upon  by  Professor  Gardner  Williams*,  and  used  to  determine 
the  distribution  of  velocities  of  the  water  flowing  in  circular  pipes. 

The  arrangement  shown  in  Fig.  146,  is  a  modified  form  of  the 
apparatus  used  by  Freeman  t  to  determine  the  distribution  of 
velocities  in  a  jet  of  water  issuing  from  a  fire  hose  under  con- 
siderable pressure.  As  shown  in  the  sketch,  the  small  orifice  0 
receives  the  impact  of  the  stream  and  two  small  holes  Q  are  drilled 
in  the  tube  T  in  a  direction  perpendicular  to  the  flow.  The  lower 
part  of  the  apparatus  OY,  as  shown  in  the  sectional  plan,  is  made 
boat-shaped  so  as  to  prevent  the  formation  of  eddies  in  the 
neighbourhood  of  the  orifices.  The  pressure  at  the  orifice  O  is 
transmitted  through  the  tube  OS,  and  the  pressure  at  Q  through 
the  tube  QR.  To  measure  the  difference  of  pressure,  or  head, 
in  the  two  tubes,  OS  and  QE.  were  connected  to  a  differential 
gauge,  similar  to  that  described  in  section  13  and  very  small 
differences  of  head  could  thus  be  obtained  with  great  accuracy. 

The  tube  shown  in  Fig.  145  has  a  cigar-shaped  bulb,  the 
impact  orifice  0  being  at  one  end  and  communicating  with  the 
tube  OS.  There  are  four  small  openings  in  the  side  of  the  bulb, 
so  that  any  variations  of  pressure  outside  are  equalised  in  the 
bulb.  The  pressures  are  transmitted  through  the  tubes  OS  and 
TR  to  a  differential  gauge  as  in  the  case  above. 

In  Fig.  147  is  shown  a  special  stuffing-box  used  by  Professor 
Williams,  to  allow  the  tube  to  be  moved  to  the  various  positions  in 

*  For  other  forms  of  Pitot  tubes  as  used  by  Professor  Williams,  E.  S.  Cole  and 
others,  see  Proceedings  of  the  Am.S.C.E.,  Vol.  xxvu. 
t  Transactions  of  the  Am.S.C.E.,  Vol.  xxi. 

16—2 


244 


HYDRAULICS 


the  cross  section  of  a  pipe,  at  which  it  was  desired  to  determine 
the  velocity  of  translation  of  the  water*. 

Mr  E.  S.  Colet  has  used  the  Pitot  tube  as  a  continuous  meter, 
the  arrangement  being  shown  in  Fig.  148.  The  tubes  were  con- 
nected to  a  U  tube  containing  a  mixture  of  carbon  tetrachloride 
and  gasoline  of  specific  gravity  1*25.  The  difference  of  level  of 
the  two  columns  was  registered  continuously  by  photography. 


To  Gauge 


Fig.  147. 


Fig.  146.  Fig.  148. 

The  tubes  shown  in  Figs.  149 — 150,  were  used  by  Bazin  to 
determine  the  distribution  of  velocity  in  the  interior  of  jets  issuing 

*  See  page  144. 

t  Proc.  A.M.S.C.E.,  Vol.  xxvu.    See  also  experiments  by  Murphy  and  Torrance 
in  same  volume. 


GAUGING  THE   FLOW  OF  WATER 


245 


from  orifices,  and  in  the  interior  of  the  nappes  of  weirs.  Each 
tube  consisted  of  a  copper  plate  1'89  inches  wide,  by  "1181  inch 
thick,  sharpened  on  the  upper  edge  and  having  two  brass  tubes 
"0787  inch  diameter,  soldered  along  the  other  edge,  and  having 
orifices  '059  inch  diameter,  0'394  inch  apart.  The  opening  in  tube 
A  was  arranged  perpendicular  to  the  stream,  and  in  B  on  the  face 
of  the  plate  parallel  to  the  stream. 


Fig.  149. 

151.     Calibration  of  Pitot  tubes. 

Whatever  the  form  of  the   Pitot  tube,  the   head   h  can  be 
expressed  as 


or 


k  being  called  the  coefficient  of  the  tube. 

This  coefficient  k  must  be  determined  by  experiment  under 
conditions  as  near  as  possible  like  those  under  which  the  tube  will 
be  used  to  determine  velocities. 

To  calibrate  the  tubes  used  in  the  determination  of  the  distri- 
bution of  velocities  in  open  channels,  Darcy  *  and  Bazin  used  three 
distinct  methods. 

(a)  The  tube  was  placed  in  front  of  a  boat  which  was  drawn 
through  still  water  at  different  velocities.  The  coefficient  was 
1'034.  This  was  considered  too  large  as  the  bow  of  the  boat 
probably  tilted  a  little,  as  it  moved  through  the  water,  thus  tilting 
the  tube  so  that  the  orifice  was  not  exactly  vertical. 

(6)  The  tube  was  placed  in  a  stream,  the  velocity  of  which 
was  determined  by  floats.  The  coefficient  was  1*006. 

(c)  Readings  were  taken  at  different  points  in  the  cross 
section  of  a  channel,  the  total  flow  Q  through  which  was  carefully 
measured  by  means  of  a  weir.  The  water  section  was  divided 

*  Eecherches  Hydrauliques. 


246  HYDRAULICS 

into  areas,  and  about  the  centre  of  each  a  reading  of  the  tube 
was  taken.  Calling  a  the  area  of  one  of  these  sections,  and  h  the 
reading  of  the  tube,  the  coefficient 


and  was  found  to  be  '993. 

Darcy*  and  Bazin  also  found  that  by  changing  the  position  of 
the  orifice  in  the  pressure  tube  the  coefficients  changed  con- 
siderably. 

Williams,  Hubbell  and  Fenkell  used  two  methods  of  calibration 
which  gave  very  different  results. 

The  first  method  was  to  move  the  tubes  through  still  water  at 
known  velocities.  For  this  purpose  a  circumferential  trough, 
rectangular  in  section,  9  inches  wide  and  8  inches  deep  was  built  of 
galvanised  iron.  The  diameter  of  its  centre  line,  which  was  made 
the  path  of  the  tube,  was  11  feet  10  inches.  The  tube  to  be  rated 
was  supported  upon  an  arm  attached  to  a  central  shaft  which  was 
free  to  revolve  in  bearings  on  the  floor  and  ceiling,  and  which  also 
supported  the  gauge  and  a  seat  for  the  observer.  The  gauge  was 
connected  with  the  tube  by  rubber  hose.  The  arm  carrying  the 
tube  was  revolved  by  a  man  walking  behind  it,  at  as  uniform  a 
rate  as  possible,  the  time  of  the  revolution  being  taken  by  means 
of  a  watch  reading  to  -5-  of  a  second.  The  velocity  was  main- 
tained as  nearly  constant  as  possible  for  at  least  a  period  of 
5  minutes.  The  value  of  k  as  determined  by  this  method  was  '926 
for  the  tube  shown  in  Fig.  145. 

In  the  second  method  adopted  by  these  workers,  the  tube  was 
inserted  into  a  brass  pipe  2  inches  in  diameter,  the  discharge 
through  which  was  obtained  by  weighing.  Readings  were  taken 
at  various  positions  on  a  diameter  of  the  pipe,  while  the  flow  in  the 
pipe  was  kept  constant.  The  values  of  */2gh,  which  may  be  called 
the  tube  velocities,  could  then  be  calculated,  and  the  mean  value 
V«,  of  them  obtained.  It  was  found  that,  in  the  cases  in  which  the 
form  of  the  tube  was  such  that  the  volume  occupied  by  it  in  the  pipe 
was  not  sufficient  to  modify  the  flow,  the  velocity  was  a  maximum 
at,  or  near,  the  centre  of  the  pipe.  Calling  this  maximum  velocity 

Vc,  the  ratio  ^p  for  a  given  set  of  readings  was  found  to  be  '81. 
Vc 

Previous  experiments  on  a  cast-iron  pipe  line  at  Detroit  having 

y 
shown  that  the  ratio  ^  was  practically  constant  for  all  velocities, 

a  similar  condition  was  assumed  to  obtain  in  the  case  of  the  brass 

*  Recherches  Hydrauliques. 


GAUGING   THE   FLOW  OF  WATER  247 

pipe.  The  tube  was  then  fixed  at  the  centre  of  the  pipe,  and 
readings  taken  for  various  rates  of  discharge,  the  mean  velocity 
U,  as  determined  by  weight,  varying  from  £  to  6  feet  per  second. 

For  the  values  of   Ji  thus  determined,  it  was  found  that     , 


was  practically  constant.     This  ratio  was  '729  for  the  tube  shown 
in  Fig.  145. 

Then  since  for  any  reading  h  of  the  tube,  the  velocity  v  is 


the  actual  mean  velocity        U  =  kVm, 

*=£• 
T~T£- 

V  m         V  c   V  m 

Therefore 

j      ratio  of  U  to  Yc  _  '729  _  ft 
ratioofVmtoVc~~'814~ 

For  the  tube  shown  in  Fig.  146,  some  of  the  values  of  k  as 
determined  by  the  two  methods  differed  very  considerably. 

Comparison  of  the  values  of  k  by  the  two  methods.  It  will  be 
seen  that  the  value  of  k  as  determined  by  moving  the  tube  through 
still  water  differs  very  considerably  from  that  obtained  in 
running  water.  In  the  latter  case  the  pressure  was  considerably 
higher  than  in  the  former,  and  it  appears  therefore,  that  k  depends 
not  only  upon  the  form  of  the  tube  but  upon  the  pressure  under 
which  it  is  working.  It  is,  clearly,  of  considerable  importance 
that  the  value  of  k  shall  be  determined  for  conditions  similar 
to  those  under  which  the  tube  is  to  be  finally  used.  This 
uncertainty  of  the  value  of  the  coefficient  under  varying  con- 
ditions of  pressure,  and  the  difficulty  in  any  case  of  accurately 
determining  it,  and  the  danger  of  its  alteration  by  objects  floating 
in  the  stream,  makes  the  use  of  the  Pitot  tube  as  a  velocity 
measurer  somewhat  uncertain,  and  it  should  be  used  with  con- 
siderable care.  In  the  hands  of  Darcy  and  Bazin  it  proved  an 
excellent  instrument  in  the  measurement  of  small  velocities  in 
open  canals,  but  for  the  determination  of  velocities  in  closed 
channels  in  which  the  pressure  is  greater,  it  does  not  seem  so 
reliable. 

152.     Gauging  by  a  weir. 

When  a  stream  is  so  small  that  a  barrier  or  dam  can  be  easily 
constructed  across  it,  or  when  a  large  quantity  of  water  is  required 
to  be  gauged  in  the  laboratory,  the  flow  can  be  determined  by 
means  of  a  notch  or  weir. 


248 


HYDRAULICS 


The  channel  as  it  approaches  the  weir  should  be  as  far  as 
possible  uniform  in  section,  and  it  is  desirable  for  accurate 
gauging*,  that  the  sides  of  the  channel  be  made  vertical,  and  the 
width  equal  to  the  width  of  the  weir.  The  sill  should  be  sharp- 
edged,  and  perfectly  horizontal,  and  as  high  as  possible  above  the 

bed  of  the  stream,  and  the  down-stream  channel 

should  be  wider  than  the  weir  to  ensure  atmospheric 
pressure  under  the  nappe.  The  difference  in  level 
of  the  sill  and  the  surface  of  the  water,  before  it 
begins  to  slope  towards  the  weir,  should  be  ac- 
curately measured.  This  is  best  done  by  a  Boyden 
hook  gauge. 

153.    The  hook  gauge. 

A  simple  form  of  hook  gauge  as  made  by  Grurley 
is  shown  in  Fig.  151.  In  a  rectangular  groove  formed 
in  a  frame  of  wood,  three  or  four  feet  long,  slides 
another  piece  of  wood  S  to  which  is  attached  a  scale 
graduated  in  feet  and  hundredths,  similar  to  a  level 
staff.  To  the  lower  end  of  the  scale  is  connected  a 
hook  H,  which  has  a  sharp  point.  At  the  upper  end 
of  the  scale  is  a  screw  T  which  passes  through  a  lug, 
connected  to  a  second  sliding  piece  L.  This  sliding 
piece  can  be  clamped  to  the  frame  in  any  position 
by  means  of  a  nut,  not  shown.  The  scale  can  then 
be  moved,  either  up  or  down,  by  means  of  the  milled 
nut.  A  vernier  Y  is  fixed  to  the  frame  by  two  small 
screws  passing  through  slot  holes,  which  allow  for  a 
slight  adjustment  of  the  zero.  At  some  point  a  few 
feet  up-stream  from  the  weir*,  the  frame  can  be 
fixed  to  a  post,  or  better  still  to  the  side  of  a  box 
from  which  a  pipe  runs  into  the  stream.  The  level 
of  the  water  in  the  box  will  thus  be  the  same  as  the 
level  in  the  stream.  The  exact  level  of  the  crest  of 
the  weir  must  be  obtained  by  means  of  a  level  and  a 
line  marked  on  the  box  at  the  same  height  as  the 
crest.  The  slider  L  can  be  moved,  so  that  the  hook 
point  is  nearly  coincident  with  the  mark,  and  the 
final  adjustment  made  by  means  of  the  screw  T. 
The  vernier  can  be  adjusted  so  that  its  zero  is 
coincident  with  the  zero  of  the  scale,  and  the  slider 
again  raised  until  the  hook  approaches  the  surface  of  the  water. 
By  means  of  the  screw,  the  hook  is  raised  slowly,  until,  by  piercing 


*  See  section  82. 


GAUGING  THE  FLOW  OF  WATER 


c 


Fig.  152.    Bazin's  Hook  Gauge. 


250 


HYDRAULICS 


the  surface  of  the  water,  it  causes  a  distortion  of  the  light  reflected 
from  the  surface.  On  moving  the  hook  downwards  again  very 
slightly,  the  exact  surface  will  be  indicated  when  the  distortion 
disappears. 

A  more  elaborate  hook  gauge,  as  used  by  Bazin  for  his  experi- 
mental work,  is  shown  in  Fig.  152. 

For  rough  gaugings  a  post  can  be  driven  into  the  bed  of  the 
channel,  a  few  feet  above  the  weir,  until  the  top  of  the  post  is 
level  with  the  sill  of  the  weir.  The  height  of  the  water  surface 


Fig.  154.    .Recording  Apparatus  Kent  Venturi  Meter. 


GAUGING  THE   FLOW  OF  WATER 


251 


above  the  top  of  the  post  can  then  be  measured  by  any  convenient 
scale. 

154.     Gauging  the  flow  in  pipes;  Venturi  meter. 

Such  methods  as  already  described  are  inapplicable  to  the 
measurement  of  the  flow  in  pipes,  in  which  it  is  necessary  that 
there  shall  be  no  discontinuity  in  the  flow,  and  special  meters  have 
accordingly  been  devised. 

For  large  pipes,  the  Yenturi  meter,  Fig.  153,  is  largely  used  in 
America,  and  is  coming  into  favour  in  this  country. 

The  theory  of  the  meter  has  already  been  discussed  (p.  44), 
and  it  was  shown  that  the  discharge  is  proportional  to  th§  square 
root  of  the  difference  H  of  the  head  at  the  throat  and  the  bend  in 
the  pipe,  or 

Q=  *^^ 

V  Oj\  —  Q," 

fc*  being  a  coefficient. 

For  measuring  the  pressure  heads  at  the  two  ends  of  the  cone, 
Mr  W.  Gr.  Kent  uses  the  arrangement  shown  in  Fig.  154. 


Fig.  155.    Recording  drum  of  the  Kent  Venturi  Meter. 
*  See  page  46. 


252 


HYDRAULICS 


The  two  pressure  tubes  from  the  meter  are  connected  to  a  U  tube 
consisting  of  two  iron  cylinders  containing  mercury.  Upon  the 
surface  of  the  mercury  in  each  cylinder  is  a  float  made  of  iron  and 
vulcanite;  these  floats  rise  or  fall  with  the  surfaces  of  the  mercury. 


Fig.  156.    Integrating  drum  of  the  Kent  Venturi  Meter. 

When  no  water  is  passing  through  the  meter,  the  mercury  in  the 
two  cylinders  stands  at  the  same  level.  When  flow  takes  place 
the  mercury  in  the  left  cylinder  rises,  and  that  in  the  right 
cylinder  is  depressed  until  the  difference  of  level  of  the  surfaces 


GAUGING  THE1  FLOW  OF  WATER 


253 


TT 

of  the  mercury  is  equal  to  — ,  s  being  the  specific  gravity  of  the 

s 

mercury  and  H  the  difference  of  pressure  head  in  the  two 
cylinders.  The  two  tubes  are  equal  in  diameter,  so  that  the  rise 
in  the  one  is  exactly  equal  to  the  fall  in  the  other,  and  the  move- 
ment of  either  rack  is  proportional  to  H.  The  discharge  is 
proportional  to  \/H,  and  arrangements  are  made  in  the  recording 
apparatus  to  make  the  revolutions  of  the  counter  proportional  to 
\/H.  To  the  floats,  inside  the  cylinders,  are  connected  racks,  as 
shown  in  Fig.  154,  gearing  with  small  pinions.  Outside  the 
mercury  cylinders  are  two  other  racks,  to  each  of  which  vertical 
motion  is  given  by  a  pinion  fixed  to  the  same  spindle  as  the  pinion 
gearing  with  the  rack  in  the  cylinder.  The  rack  outside  the  left 
cylinder  has  connected  to  it  a  light  pen  carriage,  the  pen  of  which 


Ci 


CE-nv/W 


Fig.  157.     Kent  Venturi  Meter.     Development  of  Integrating  drum. 

makes  a  continuous  record  on  the  diagram  drum  shown  in 
Fig.  155.  This  drum  is  rotated  at  a  uniform  rate  by  clockwork, 
and  on  suitably  prepared  paper  a  curve  showing  the  rate  of 
discharge  at  any  instant  is  thus  recorded.  The  rack  outside  the 
right  cylinder  is  connected  to  a  carriage,  the  function  of  which  is 
to  regulate  the  rotations  of  the  counter  which  records  the  total 
flow.  Concentric  with  the  diagram  drum  shown  in  Fig.  155,  and 
within  it,  is  a  second  drum,  shown  in  Fig.  156,  which  also  rotates 
at  a  uniform  rate.  Fig.  157  shows  this  internal  drum  developed. 
The  surface  of  the  drum  below  the  parabolic  curve  FEG-  is  recessed. 
If  the  right-hand  carriage  is  touching  the  drum  on  the  recessed 


254 


HYDRAULICS 


portion,  the  counter  gearing  is  in  action,  but  is  put  out  of  action 
when  the  carriage  touches  the  cylinder  on  the  raised  portion 
above  FGL  Suppose  the  mercury  in  the  right  cylinder  to  fall  a 
height  proportional  to  H,  then  the  carriage  will  be  in  contact 
with  the  drum,  as  the  drum  rotates,  along  the  line  CD,  but  the 
recorder  will  only  be  in  operation  while  the  carriage  is  in 
contact  along  the  length  CE.  Since  FGr  is  a  parabolic  curve  the 
fraction  of  the  circumference  CE  =  ra .  \/H,  m  being  a  constant, 
and  therefore  for  any  displacement  H  of  the  floats  the  counter  for 
each  revolution  of  the  drum  will  be  in  action  for  a  period  propor- 
tional to  \/H.  When  the  float  is  at  the  top  of  the  right  cylinder, 
the  carriage  is  at  the  top  of  the  drum,  and  in  contact  with  the 
raised  portion  for  the  whole  of  a  revolution  and  no  flow  is 
registered.  When  the  right  float  is  in  its  lowest  position  the 
carriage  is  at  the  bottom  of  the  drum,  and  flow  is  registered 
during  the  whole  of  a  revolution.  The  recording  apparatus  can 
be  placed  at  any  convenient  distance  less  than  1000  feet  from 
the  meter,  the  connecting  tubes  being  made  larger  as  the  distance 
is  increased. 

155.    Deacon's  waste-water  meter. 

An  ingenious  and  very  simple  meter  designed  by  Mr  Gr.  F. 
Deacon  principally  for  detecting  the  leakage  of  water  from  pipes 
is  as  shown  in  Fig.  158. 


Fig.  158.     Deacon  waste-water  meter. 

The  body  of  the  meter  which  is  made  of  cast-iron,  has  fitted 
into  it  a  hollow  cone  C  made  of  brass.  A  disc  D  of  the  same  diameter 
as  the  upper  end  of  the  cone  is  suspended  in  this  cone  by  means  of 
a  fine  wire,  which  passes  over  a  pulley  not  shown ;  the  other  end 
of  the  wire  carries  a  balance  weight. 


GAUGING  THE   FLOW  OF  WATER  255 

When  no  water  passes  through  the  meter  the  disc  is  drawn  to 
the  top  of  the  cone,  but  when  water  is  drawn  through,  the  disc  is 
pressed  downwards  to  a  position  depending  upon  the  quantity  of 
water  passing.  A  pencil  is  attached  to  the  wire,  and  the  motion 
of  the  disc  can  then  be  recorded  upon  a  drum  made  to  revolve  by 
clockwork.  The  position  of  the  pencil  indicates  the  rate  of  flow 
passing  through  the  meter  at  any  instant. 

When  used  as  a  waste-water  meter,  it  is  placed  in  a  by-pass 
leading  from  the  main,  as  shown  diagrammatically  in  Fig.  159. 


B 

'S.V 

A 

xS.Y        D 

Fig.  159. 

The  valves  A  and  B  are  closed  and  the  valve  C  opened.  The 
rate  of  consumption  in  the  pipe  AD  at  those  hours  of  the  night 
when  the  actual  consumption  is  very  small,  can  thus  be  deter- 
mined, and  an  estimate  made  as  to  the  probable  amount  wasted. 

If  waste  is  taking  place,  a  careful  inspection  of  the  district 
supplied  by  the  main  AD  may  then  be  made  to  detect  where  the 
waste  is  occurring. 

156.     Kennedy's  meter. 

This  is  a  positive  meter  in  which  the  volume  of  water  passing 
through  the  meter  is  measured  by  the  displacement  of  a  piston 
working  in  the  measuring  cylinder. 

The  long  hollow  piston  P,  Fig.  157,  fits  loosely  in  the  cylinder 
Co,  but  is  made  water-tight  by  means  of  a  cylindrical  ring  of 
rubber  which  rolls  between  the  piston  and  the  inside  of  the 
cylinder,  the  friction  being  thus  reduced  to  a  minimum.  At  each 
end  of  the  cylinder  is  a  rubber  ring,  which  makes  a  water-tight 
joint  when  the  piston  is  forced  to  either  end  of  the  cylinder,  so 
that  the  rubber  roller  has  only  to  make  a  joint  while  the  piston  is 
free  to  move. 

The  water  enters  the  meter  at  A,  Fig.  1616,  and  for  the 
position  shown  of  the  regulating  cock,  it  flows  down  the  passage 
D  and  under  the  piston.  The  piston  rises,  and  as  it  does  so  the 
rack  R  turns  the  pinion  S,  and  thus  the  pinion  p  which  is  keyed 
to  the  same  spindle  as  S.  This  spindle  also  carries  loosely 
a  weighted  lever  W,  which  is  moved  as  the  spindle  revolves  by 
either  of  two  projecting  fingers.  As  the  piston  continues  to 
ascend,  the  weighted  lever  is  moved  by  one  of  the  fingers  until  its 


256 


HYDRAULICS 


centre  of  gravity  passes  the  vertical  position,  when  it  suddenly 
falls  on  to  a  buffer,  and  in  its  motion  moves  the  lever  L,  which 
turns  the  cock,  Fig.  161  b,  into  a  position  at  right  angles  to  that 


Rubber  Seating 


-Rubber  Rolling 
Packing 


Rubber  Seating 


Fig.  160. 


GAUGING  THE  FLOW  OF  WATER 


257 


shown.     The  water  now  passes  from  A  through  the  passage  C, 
and  thus  to  the  top  of  the  cylinder,  and  as  the  piston  descends, 


Fig.  161  a. 


Fig.  161  &. 


L.    H. 


17 


258 


HYDRAULICS 


the  water  that  is  below  it  passes  to  the  outlet  B.  The  motion  of 
the  pinion  S  is  now  reversed,  and  the  weight  W  lifted  until  it 
again  reaches  the  vertical  position,  when  it  falls,  bringing  the 
cock  C  into  the  position  shown  in  the  figure,  and  another  up 


S 


Fig.  161  c. 

stroke  is  commenced.  The  oscillations  of  the  pinion  p  are  trans- 
ferred to  the  counter  mechanism  through  the  pinions  pi  and  p2, 
Fig.  161  a,  in  each  of  which  is  a  ratchet  and  pawl.  The  counter 
is  thus  rotated  in  the  same  direction  whichever  way  the  piston 
moves. 

157.     Gauging  the  flow  of  streams  by  chemical  means. 

Mr  Stromeyer*  has  very  successfully  gauged  the  quantity  of 
water  supplied  to  boilers,  and  also 
the  flow  of  streams  by  mixing 
with  the  stream  during  a  definite 
time  and  at  a  uniform  rate,  a 
known  quantity  of  a  concentrated 
solution  of  some  chemical,  the 
presence  of  which  in  water,  even 
in  very  small  quantities,  can  be 
easily  detected  by  some  sensitive 
reagent.  Suppose  for  instance 
water  is  flowing  along  a  small 
stream.  Two  stations  at  a  known 
distance  apart  are  taken,  and  the 
time  determined  which  it  takes 
the  water  to  traverse  the  dis- 
tance between  them.  At  a  stated 
time,  by  means  of  a  special  ap- 
paratus— Mr  Stromeyer  uses  the 
arrangement  shown  in  Fig.  162 
— sulphuric  acid,  say,  of  known 
strength,  is  run  into  the  stream  at  a  known  rate,  at  the  upper 


Jv 

i 


Fig.  162. 


*  Transactions  of  Naval  Architects,  1896  ;  Proceedings  Inst.  C.E.,  Vol.  CLX. 


GAUGING  THE  FLOW  OF  WATER  259 

station.  While  the  acid  is  being  put  into  the  stream,  a  small 
distance  up-stream  from  where  the  acid  is  introduced  samples  of 
water  are  taken  at  definite  intervals.  At  the  lower  station 
sampling  is  commenced,  at  a  time,  after  the  insertion  of  the 
acid  at  the  upper  station  is  started,  equal  to  that  required  by  the 
water  to  traverse  the  distance  between  the  stations,  and  samples 
are  then  taken,  at  the  same  intervals,  as  at  the  upper  station. 
The  quantity  of  acid  in  a  known  volume  of  the  samples  taken 
at  the  upper  and  lower  station  is  then  determined  by  analysis. 
In  a  volume  V0  of  the  samples,  let  the  difference  in  the  amount  of 
sulphuric  acid  be  equivalent  to  a  volume  v0  of  pure  sulphuric 
acid.  If  in  a  time  t,  a  volume  Y  of  water,  has  flowed  down  the 
stream,  and  there  has  been  mixed  with  this  a  volume  v  of  pure 
sulphuric  acid,  then,  if  the  acid  has  mixed  uniformly  with  the 
water,  the  ratio  of  the  quantity  of  water  flowing  down  the  stream 
to  the  quantity  of  acid  put  into  the  stream,  is  the  same  as  the 
ratio  of  the  volume  of  the  sample  tested  to  the  difference  of  the 
volume  of  the  acid  in  the  samples  at  the  two  stations,  or 

V  =  V0 

V         V0  ' 

Mr  Stromeyer  considers  that  the  flow  in  the  largest  rivers  can 
be  determined  by  this  method  within  one  per  cent,  of  its  true  value. 

In  large  streams  special  precautions  have  to  be  taken  in 
putting  the  chemical  solution  into  the  water,  to  ensure  a  uniform 
mixture,  and  also  special  precautions  must  be  adopted  in  taking 
samples. 

For  other  important  information  upon  this  interesting  method 
of  measuring  the  flow  of  water  the  reader  is  referred  to  the  two 
papers  cited  above. 

An  apparatus  for  accurately  gauging  the  flow  of  the  solution 
is  shown  in  Fig.  162.  The  chemical  solution  is  delivered  into 
a  cylindrical  tank  by  means  of  a  pipe  I.  On  the  surface  of  the 
solution  floats  a  cork  which  carries  a  siphon  pipe  SS,  and  a  balance 
weight  to  keep  the  cork  horizontal.  After  the  flow  has  been 
commenced,  the  head  h  above  the  orifice  is  clearly  maintained 
constant,  whatever  the  level  of  the  surface  of  the  solution  in  the 
tank. 


17—2 


260 


HYDRAULICS 


EXAMPLES. 

(1)  Some  observations  are  made  by  towing  a  current  meter,  with  the 
following  results : — 

Speed  in  ft.  per  sec.  Bevs.  of  meter  per  min. 
1  80 

5  560 

Find  an  equation  for  the  meter. 

(2)  Describe  two  methods  of  gauging  a  large  river,  from  observations 
in  vertical  and  horizontal  planes;   and  state  the  nature  of  the  results 
obtained. 

If  the  cross  section  of  a  river  is  known,  explain  how  the  approximate 
discharge  may  be  estimated  by  observation  of  the  mid-surface  velocity 
alone. 

(3)  The  following  observations  of  head  and  the  corresponding  discharge 
were  made  in  connection  with  a  weir  6'53  feet  wide. 


Head  in  feet       O'l 

Discharge  in  cubic  feet  per 


sec.  per  foot  width 


0-17 


0-5 
1-2 


1-0 


3-35 


1-5 


6-1 


2-0 
9-32 


2-5 

13-03 


3-0 
17-03 


3-5 


21-54 


4-0 


26-4 


Assuming  the  law  connecting  the  head  h  with  the  discharge  Q  as 


find  m  and  n.     (Plot  logarithms  of  Q  and  h.) 

(4)  The  following  values  of  Q  and  h  were  obtained  for  a  sharp  -edge 
weir  6'53  feet  long,  without  lateral  contraction.  Find  the  coefficient  of 
discharge  at  various  heads. 


Head  ft  ...   -1   1-4 
Q  per  foot- 
length  ...   -17  | -87 


•6   |    -8 
1-56  2-37 


1-0 
3-35 


2-0 
9-32 


2-5 
13-03 


3-0 
17-03 


3-5 
21-54 


4-0 
26-4 


4-5 
31-62 


5-0 
37-09 


5-5 
42-81 


(5)  The  following  values  of  the  head  over  a  weir  10  feet  long  were 
obtained  at  5  minutes  intervals. 

Head  in  feet          '35     -36     '37     '37     '38    -39     -40    -41     -42     -40     -39     -41 
Taking  the  coefficient  of  discharge  C  as  3'36,  find  the  discharge  in 
one  hour. 

(6)  A  Pitot  tube  was  calibrated  by  moving  it  through  still  water  in  a 
tank,  the  tube  being  fixed  to   an  arm  which  was  made  to  revolve  at 
constant  speed  about  a  fixed  centre.     The  following  were  the  velocities  of 
the  tube  and  the  heads  measured  in  inches  of  water. 


Velocities  ft.  per  sec.    1-432 
Head  in  inches 

of  water  -448 


1-738 


•663 


2-275 
1-02 


2-713 


1-69 


3-235 
2-07 


3-873 

2-88 


4-983 
5-40 


5-584 
6-97 


6-142 
8-51 


Determine  the  coefficient  of  the  tube. 

For  examples  on  Venturi  meters  see  Chapter  II. 


CHAPTER  VIII. 


IMPACT   OF  WATER   ON  VANES. 

158.  Definition  of  a  vector.  A  right  line  AB,  considered  as 
having  not  only  length,  but  also  direction,  and  sense,  is  said  to  be 
a  vector*.  The  initial  point  A  is  said  to  be  the  origin. 

It  is  important  that  the  difference  between  sense  and  direction 
should  be  clearly  recognised. 

Suppose  for  example,  from  any  point  A,  a  line  AB  of 
definite  length  is  drawn  in  a  northerly  direction,  then  the 
direction  of  the  line  is  either  from  south  to  north  or  north  to 
south,  but  the  sense  of  the  vector  is  definite,  and  is  from  A  to  B, 
that  is  from  south  to  north. 

The  vector  AB  is  equal  in  magnitude  to  the  vector  BA,  but 
they  are  of  opposite  sign  or, 

AB  =  -BA. 

The  sense  of  the  vector  is  indicated  by  an  arrow,  as  on  AB, 
Fig.  163. 

Any  quantity  which  has  magnitude,  direction,  and  sense,  may 
be  represented  by  a  vector. 


B 


Fig.  163. 


For  example,  a  body  is  moving  with  a  given  velocity  in  a 
given  direction,  sense  being  now  implied.  Then  a  line  AB  drawn 
parallel  to  the  direction  of  motion,  and  on  some  scale  equal  in 

*  Sir  W.  Hamilton,  Quaternions. 


262 


HYDRAULICS 


length  to  the  velocity  of  the  body  is  the  velocity  vector ;  the  sense 
is  from  A  to  B. 

159.  *  Sum  of  two  vectors. 

If  a  and  ft  Fig.  163,  are  two  vectors  the  sum  of  these  vectors 
is  found,  by  drawing  the  vectors,  so  that  the  beginning  of  ft  is  at 
the  end  of  a,  and  joining  the  beginning  of  a  to  the  end  of  ft 
Thus  7  is  the  vector  sum  of  a  and  ft. 

160.  Resultant  of  two  velocities. 

When  a  body  has  impressed  upon  it  at  any  instant  two 
velocities,  the  resultant  velocity  of  the  body  in  magnitude  and 
direction  is  the  vector  sum  of  the  two  impressed  velocities.  This 
may  be  stated  in  a  way  that  is  more  definitely  applicable  to  the 
problems  to  be  hereafter  dealt  with,  as  follows.  If  a  body  is 
moving  with  a  given  velocity  in  a  given  direction,  and  a  second 
velocity  is  impressed  upon  the  body,  the  resultant  velocity  is  the 
vector  sum  of  the  initial  and  impressed  velocities. 

Example.  Suppose  a  particle  of  water  to  be  moving  along  a  vane  DA,  Fig.  164, 
with  a  velocity  Vr,  relative  to  the  vane. 

If  the  vane  is  at  rest,  the  particle  will  leave  it  at  A  with  this  velocity. 

If  the  vane  is  made  to  move  in  the  direction  EF  with  a  velocity  v,  and  the 
particle  has  still  a  velocity  Vr  relative  to  the  vane,  and  remains  in  contact  with  the 
vane  until  the  point  A  is  reached,  the  velocity  of  the  water  as  it  leaves  the  vane  at 
A,  will  be  the  vector  sum  y  of  a  and  /3,  i.e.  of  Vr  and  v,  or  is  equal  to  u. 

161.  Difference  of  two  vectors. 

The  difference  of  two  vectors  a  and  ft  is  found  by  drawing  both 
vectors  from  a  common  origin  A,  and  joining  the  end  of  ft  to  the 
end  of  a.  Thus,  CB,  Fig.  165,  is  the  difference  of  the  two  vectors 
a  and  ft  or  y  =  a  —  ft  and  BC  is  equal  to  ft  -  a,  or  ft  -  a  =  -  y. 


162.    Absolute  velocity. 

By  the  terms  "  absolute  velocity "  or  "  velocity "  without  the 
adjective,  as  used  in  this  chapter,  it  should  be  clearly  understood, 
is  meant  the  velocity  of  the  moving  water  relative  to  the  earth,  or 
to  the  fixed  part  of  any  machine  in  which  the  water  is  moving. 

*  Henrici  and  Turner,  Vectors  and  Rotors. 


IMPACT   OF   WATER   ON    VANES  263 

To  avoid  repetition  of  the  word  absolute,  the  adjective  is 
frequently  dropped  and  "  velocity  "  only  is  used. 

163.  When  a  body  is  moving  with  a  velocity  U,  Fig.  166,  in 
any  direction,  and  has  its  velocity  changed  to  U'  in  any  other 
direction,  by  an  impressed  force,  the  change  in  velocity,  or  the 
velocity  that  is  impressed  on  the  body,  is  the  vector  difference  of 
the  final  and  the  initial  velocities.     If  AB  is  U,  and  AC,  U',  the 
impressed  velocity  is  BC. 

By  Newton's  second  law  of  motion,  the  resultant  impressed 
force  is  in  the  direction  of  the  change  of  velocity,  and  if  W  is  the 
weight  of  the  body  in  pounds  and  t  is  the  time  taken  to  change 
the  velocity,  the  magnitude  of  the  impressed  force  is 

W 

P  =  —  (change  of  velocity)  Ibs. 
Qt 

This  may  be  stated  more  generally  as  follows. 
The  rate  of  change  of  momentum,  in  any  direction,  is  equal  to 
the  impressed  force  in  that  direction,  or 

W  dt?,, 
P  =  — .  JT  Ibs. 
g     at 

In  hydraulic  machine  problems,  it  is  generally  only  necessary 
to  consider  the  change  of  momentum  of  the  mass  of  water  that 
acts  upon  the  machine  per  second.  W  in  the  above  equation  then 
becomes  the  weight  of  water  per  second,  and  t  being  one  second, 

W 

P  =  -  -  (change  of  velocity). 

164.  Impulse  of  water  on  vanes. 

It  follows  that  when  water  strikes  a  vane  which  is  either 
moving  or  at  rest,  and  has  its  velocity  changed,  either  in  magni- 
tude or  direction,  pressure  is  exerted  on  the  vane. 

As  an  example,  suppose  in  one  second  a  mass  of  water,  weighing 
W  Ibs.  and  moving  with  a  velocity  U  feet  per  second,  strikes  a 
fixed  vane  AD,  and  let  it  glide  upon  the  vane  at  A,  Fig.  167,  and 
leave  at  D  in  a  direction  at  right  angles  to  its  original  direction 
of  motion.  The  velocity  of  the  water  is  altered  in  direction  but 
not  in  magnitude,  the  original  velocity  being  changed  to  a  velocity 
at  right  angles  to  it  by  the  impressed  force  the  vane  exerts  upon 
the  water. 

The   change   of  velocity  in   the   direction   AC  is,   therefore, 

W 

equal  to  U,  and  the  change  of  momentum  per  second  is  — .  U 

foot  Ibs. 


264 


HYDRAULICS 


Since  W  Ibs.  of  water  strike  the  vane  per  second,  the  pressure 
P,  acting  in  the  direction  CA,  required  to  hold  the  vane  in  position 
is,  therefore, 


Fig.  167. 

Again,  the  vane  has  impressed  upon  the  water  a  velocity  U  in 
the  direction  DF  which  it  originally  did  not  possess. 
The  pressure  PI  in  the  direction  DF  is,  therefore, 

W 

P1  =  P  =  -11.TJ. 
9 

The  resultant  reaction  of  the  vane  in  magnitude  and  direction 
is,  therefore,  R  the  resultant  of  P  and  Pa . 

This  resultant  force   could    have   been 
found    at   once   by  finding   the    resultant     c 
change  in  velocity.     Set  out  ac,  Fig.  168, 
equal  to  the  initial  velocity  in  magnitude 
and  direction,  and   ad  equal   to  the  final     »x 
velocity.     The   change   in  velocity  is    the 
vector  difference  cd,  or   cd  is  the  velocity 
that  must  be   impressed   on   a  particle   of 
water   to   change  its  velocity  from   ac  to 
ad. 


Fig.  168. 


The    impressed  velocity   cd    is   Y  =  \/U2  +  U2,   and    the   total 
impressed  force  is 


IMPACT   OF  WATER  ON   VANES  265 

It  at  once  follows,  that  if  a  jet  of  water  strikes  a  fixed  plane 
perpendicularly,  with  a  velocity  U,  and  glides  along  the  plane,  the 

W 

normal  pressure  on  the  plane  is  —  U. 

y 

Example.  A  stream  of  water  1  sq.  foot  in  section  and  having  a  velocity  of 
10  feet  per  second  glides  on  to  a  fixed  vane  in  a  direction  making  an  angle  of 
30  degrees  with  a  given  direction  AB. 

The  vane  turns  the  jet  through  an  angle  of  90  degrees. 

Find  the  pressure  on  the  vane  in  the  direction  parallel  to  AB  and  the  resultant 
pressure  on  the  vane. 

In  Fig.  167,  AC  is  the  original  direction  of  the  jet  and  DF  the  final  direction. 
The  vane  simply  changes  the  direction  of  the  water,  the  final  velocity  being  equal 
to  the  initial  velocity. 

The  vector  triangle  is  acd,  Fig.  168,  ac  and  ad  being  equal. 

The  change  of  velocity  in  magnitude  and  direction  is  cd,  the  vector  difference  of 
ad  and  ac  ;  resolving  cd  parallel  to,  and  perpendicular  to  AB,  ce  is  the  change  of 
velocity  parallel  to  AB. 

Scaling  off  ce  and  calling  it  va  ,  the  force  to  be  applied  along  B  A  to  keep  the 
vane  at  rest  is, 


But  cd=*j2.10 

and  ce  =  cdcos!5° 

=  J%  .  10  .0-9659  ; 

10  x  62-4     . 

therefore,  PBA  =  —  0  .  0      x  13> 

oa'a 

=  2641bs. 
The  pressure  normal  to  AB  is 


9 
=  —  V2.10sinl5°=721bs. 


The  resultant  is 


165.    Relative  velocity. 

Before  going  on  to  the  consideration  of  moving  vanes  it  is 
important  that  the  student  should  have  clear  ideas  as  to  what  is 
meant  by  relative  velocity. 

A  train  is  said  to  have  a  velocity  of  sixty  miles  an  hour  when, 
if  it  continued  in  a  straight  line  at  a  constant  velocity  for  one 
hour,  it  would  travel  sixty  miles.  What  is  meant  is  that  the  train 
is  moving  at  sixty  miles  an  hour  relative  to  the  earth. 

If  two  trains  run  on  parallel  lines  in  the  same  direction,  one 
at  sixty  and  the  other  at  forty  miles  an  hour,  they  have  a 
relative  velocity  to  each  other  of  20  miles  an  hour.  If  they  move 
in  opposite  directions,  they  have  a  relative  velocity  of  100  miles 
an  hour.  If  one  of  the  trains  T  is  travelling  in  the  direction  AB, 
Fig.  169,  and  the  other  Ta  in  the  direction  AC,  and  it  be  supposed 
that  the  lines  on  which  they  are  travelling  cross  each  other  at  A, 


266 


HYDRAULICS 


and  the  trains  are  at  any  instant  over  each  other  at  A,  at  the  end 

of  one  minute  the  two  trains  will  be  at  B  and  C  respectively,  at 

distances  of  one  mile  and  two-thirds  of  a 

mile  from  A.     Relatively  to  the  train  T 

moving  along  AB,  the  train  TI  moving 

along  AC  has,  therefore,  a  velocity  equal 

to  BC,  in  magnitude  and  direction,  and 

relatively  to  the  train  Ta  the  train  T  has 

a  velocity  equal  to  CB.     But  AB  and  AC    « 

may  be  taken  as  the  vectors  of  the  two 

velocities,  and  BC  is  the  vector  difference 

of  AC  and  AB,  that  is,  the  velocity  of  T 

vector  difference  of  AC  and  AB. 


T, 

Fig.  169. 
relative  to  T  is  the 


166.  Definition  of  relative  velocity  as  a  vector. 

If  two  bodies  A  and  B  are  moving  with  given  velocities  v  and 
Vi  in  given  directions,  the  relative  velocity  of  A  to  B  is  the  vector 
difference  of  the  velocities  v  and  Vi. 

Thus  when  a  stream  of  water  strikes  a  moving  vane  the 
magnitude  and  direction  of  the  relative  velocity  of  the  water  and 
the  vane  is  the  vector  difference  of  the  velocity  of  the  water  and 
the  edge  of  the  vane  where  the  water  meets  it. 

167.  To  find  the  pressure  on  a  moving  vane,   and  the 
rate  of  doing  work. 

A  jet  of  water  having  a  velocity  U  strikes  a  flat  vane,  the 
plane  of  which  is  perpendicular  to  the  direction  of  the  jet,  and 
which  is  moving  in  the  same  direction  as  the  jet  with  a  velocity  v. 


..*__ 
Fig.  170 


Fig.  171. 


The  relative  velocity  of  the  water  and  the  vane  is  \J  —  v,  the 
vector  difference  of  U  and  v,  Fig.  170.  If  the  water  as  it  strikes 
the  vane  is  supposed  to  glide  along  it  as  in  Fig.  171,  it  will  do 


IMPACT   OF   WATER   ON    VANES  267 

so  with  a  velocity  equal  to  (U  —  1?),  and  as  it  moves  with  the  vane 
it  will  still  have  a  velocity  v  in  the  direction  of  motion  of  the 
vane.  Instead  of  the  water  gliding  along  the  vane,  the  velocity 
~U-v  may  be  destroyed  by  eddy  motions,  but  the  water  will  still 
have  a  velocity  v  in  the  direction  of  the  vane.  The  change  in 
velocity  in  the  direction  of  motion  is,  therefore,  the  relative 
velocity  U-v,  Fig.  170. 

For  every  pound  of  water  striking  the  vane,  the  horizontal 

U—  1» 

change  in  momentum  is  -  ,  and  this  equals  the  normal  pressure 

y 

P  on  the  vane,  per  pound  of  water  striking  the  vane. 
The  work  done  per  second  per  pound  is 


g 

The  original  kinetic  energy  of   the  jet  per  pound  of  water 

IT2 

striking  the  vane  is  ~—  ,  and  the  efficiency  of  the  vane  is,  therefore, 

_<2v(U-v) 

U2 

which  is  a  maximum  when  v  is  JU,  and  e  =  \.    An  application  of 
such  vanes  is  illustrated  in  Fig.  185,  page  292. 

Nozzle  and  single  vane.  Let  the  water  striking  a  vane  issue 
from  a  nozzle  of  area  a,  and  suppose  that  there  is  only  one  vane. 

Let  the  vane  at  a  given  instant  be  supposed  at  A,  Fig.  172.  At 
the  end  of  one  second  the  front  of  the  jet,  if  perfectly  free  to 
move,  would  have  arrived  at  B  and  the  vane  at  C.  Of  the  water 
that  has  issued  from  the  jet,  therefore,  only  the  quantity  BC  will 
have  hit  the  vane. 


Fig.  172. 
The  discharge  from  the  nozzle  is 


and  the  weight  that  hits  the  vane  per  second  is 


U 

The  change  of  momentum  per  second  is 


U 


268  HYDKAULICS 

and  the  work  done  is,  therefore, 


Or  the  work  done  per  Ib.  of  water  issuing  from  the  nozzle  is 


This  is  purely  a  hypothetical  case  and  has  no  practical 
importance. 

Nozzle  and  a  number  of  vanes.  If  there  are  a  number  of 
vanes  closely  following  each  other,  the  whole  of  the  water  issuing 
from  the  nozzle  hits  the  vanes,  and  the  work  done  is 

W(U-v)v 


The  efficiency  is 


9 

2v  (U  -  v) 
U2 


and  the  maximum  efficiency  is  -|. 

It  follows  that  an  impulse  water  wheel,  with  radial  blades,  as 
in  Fig.  185,  cannot  have  an  efficiency  of  more  than  50  per  cent. 

168.    Impact  of  water  on  a  vane  when  the  directions  of 
motion  of  the  vane  and  jet  are  not  parallel. 

Let  U  be  the  velocity  of  a  jet  of  water  and  AB  its  direction, 
Fig.  173. 

A, 


B 


Fig.  173. 

Let  the  edge  A  of  the  vane  AC  be  moving  with  a  velocity  v ; 
the  relative  velocity  Yr  of  the  water  and  the  vane  at  A  is  DB. 
From  the  triangle  DAB  it  is  seen  that,  the  vector  sum  of  the 
velocity  of  the  vane  and  the  relative  velocity  of  the  jet  and  the 
vane  is  equal  to  the  velocity  of  the  jet ;  for  clearly  U  is  the  vector 
sum  of  v  and  Vr. 

If  the  direction  of  the  tip  of  the  vane  at  A  is  made  parallel  to 
DB  the  water  will  glide  on  to  the  vane  in  exactly  the  same  way 


IMPACT   OF   WATER   ON   VANES  2C9 

as  if  it  were  at  rest,  and  the  water  were  moving  in  the  direction 
DB.  This  is  the  condition  that  no  energy  shall  be  lost  by  shock. 

When  the  water  leaves  the  vane,  the  relative  velocity  of  the 
water  and  the  vane  must  be  parallel  to  the  direction  of  the 
tangent  to  the  vane  at  the  point  where  it  leaves,  and  it  is  equal  to 
the  vector  difference  of  the  absolute  velocity  of  the  water,  and 
the  vane.  Or  the  absolute  velocity  with  which  the  water  leaves 
the  vane  is  the  vector  sum  of  the  velocity  of  the  tip  of  the  vane 
and  the  relative  velocity  of  the  water  to  the  vane. 

Let  CG-  be  the  direction  of  the  tangent  to  the  vane  at  C.  Let 
CE  be  Vi,  the  velocity  of  C  in  magnitude  and  direction,  and  let  OF 
be  the  absolute  velocity  Ui  with  which  the  water  leaves  the  vane. 

Draw  EF  parallel  to  CGr  to  meet  the  direction  OF  in  F,  then 
the  relative  velocity  of  the  water  and  the  vane  is  EF,  and  the 
velocity  with  which  the  water  leaves  the  vane  is  equal  to  CF. 

If  Vi  and  the  direction  CGr  are  given,  and  the  direction  in  which 
the  water  leaves  the  vane  is  given,  the  triangle  CEF  can  be 
drawn,  and  CF  determined. 

If  on  the  other  hand  vl  is  given,  and  the  relative  velocity  vr  is 
given  in  magnitude  and  direction,  CF  can  be  found  by  measuring 
off  along  EF  the  known  relative  velocity  vr  and  joining  CF. 

If  Vi  and  Ui  are  given,  the  direction  of  the  tangent  to  the  vane 
is  then,  as  at  inlet,  the  vector  difference  of  Ui  and  Vi . 

It  will  be  seen  that  when  the  water  either  strikes  or  leaves  the 
vane,  the  relative  velocity  of  the  water  and  the  vane  is  the  vector 
difference  of  the  velocity  of  the  water  and  the  vane,  and  the  actual 
velocity  of  the  water  as  it  leaves  the  vane  is  the  vector  sum  of  the 
velocity  of  the  vane  and  the  relative  velocity  of  the  water  and 
the  vane. 

Example.  The  direction  of  the  tip  of  the  vane  at  the  outer  circumference  of  a 
wheel  fitted  with  vanes,  makes  an  angle  of  165  degrees  with  the  direction  of  motion 
of  the  tip  of  the  vane. 

The  velocity  of  the  tip  at  the  outer  circumference  is  82  feet  per  second. 

The  water  leaves  the  wheel  in  such  a  direction  and  with  such  a  velocity  that  the 
radial  component  is  13  feet  per  second. 

Find  the  absolute  velocity  of  the  water  in  direction  and  magnitude  and  the 
relative  velocity  of  the  water  and  the  wheel. 

To  draw  the  triangle  of  velocities,  set  out  AB  equal  to  82  feet,  and  make  the 
angle  ABC  equal  to  15  degrees.  BC  is  then  parallel  to  the  tip  of  the  vane. 

Draw  EC  parallel  to  AB,  and  at  a  distance  from  it  equal  to  13  feet  and 
intersecting  BC  in  C. 

Then  AC  is  the  vector  sum  of  AB  and  BC,  and  is  the  absolute  velocity  of  the 
water  in  direction  and  magnitude. 

Expressed  trigonometrically 

AC2=(82-13cotl5°)2  +  132 

=  33-58  +  132  and  AC  =  36-7  ft.  per  sec. 

sinBAC=i?=-354. 
AU 

Therefore  BAG  =  20°  45'. 


270 


HYDRAULICS 


169.  Conditions  which  the  vanes  of  hydraulic  machines 
should  satisfy. 

In  all  properly  designed  hydraulic  machines,  such  as  turbines, 
water  wheels,  and  centrifugal  pumps,  in  which  water  flowing  in 
a  definite  direction  impinges  on  moving  vanes,  the  relative  velocity 
of  the  water  and  the  vanes  should  be  parallel  to  the  direction  of 
the  vanes  at  the  point  of  contact.  If  not,  the  water  breaks  into 
eddies  as  it  moves  on  to  the  vanes  and  energy  is  lost. 

Again,  if  in  such  machines  the  water  is  required  to  leave  the 
vanes  with  a  given  velocity  in  magnitude  and  direction,  it  is  only 
necessary  to  make  the  tip  of  the  vane  parallel  to  the  vector 
difference  of  the  given  velocity  with  which  the  water  is  to  leave 
the  vane  and  the  velocity  of  the  tip  of  the  vane. 

Example  (1).  A  jet  of  water,  Fig.  174,  moves  in  a  direction  AB  making  an  angle 
of  30  degrees  with  the  direction  of  motion  AC  of  a  vane  moving  in  the  atmosphere. 
The  jet  has  a  velocity  of  30  ft.  per  second  and  the  vane  of  15  ft.  per  second.  To  find 
(a)  the  direction  of  the  vane  at  A  so  that  the  water  may  enter  without  shock;  (&)  the 
direction  of  the  tangent  to  the  vane  where  the  water  leaves  it,  so  that  the  absolute 
velocity  of  the  water  when  it  leaves  the  vane  is  in  a  direction  perpendicular  to  AC ; 
(c)  the  pressure  on  the  vane  and  the  work  done  per  second  per  pound  of  water 
striking  the  vane.  Friction  is  neglected. 


u, 


CFiange>  oCVeLodty  in/  the 
ofmotiori. 


Fig.  174. 

The  relative  velocity  Vr  of  the  water  and  the  vane  at  A  is  CB,  and  for  no  shock 
the  vane  at  A  must  be  parallel  to  CB. 

Since  there  is  no  friction,  the  relative  velocity  Vr  of  the  water  and  the  vane 
cannot  alter,  and  therefore,  the  triangle  of  velocities  at  exit  is  ACD  or  FAjCj . 

The  point  D  is  found,  by  taking  C  as  centre  and  CB  as  radius  and  striking  the 
arc  BD  to  cut  the  known  direction  AD  in  D. 

The  total  change  of  velocity  of  the  jet  is  the  vector  difference  DB  of  the  initial 
and  final  velocities,  and  the  change  of  velocity  in  the  direction  of  motion  is  BE. 
Calling  this  velocity  V,  the  pressure  exerted  upon  the  vane  in  the  direction  of 
motion  is 

Y 

—  Ibs.  per  Ib.  of  water  striking  the  vane. 

y 

The  work  done  per  Ib.  is,  therefore,  —  ft.  Ibs.  and  the  efficiency,  since  there  is 

no  loss  by  friction,  or  shock,  is 

Vv        2Vt; 

— T5*-1F' 

9?r, 


IMPACT   OF   WATER   ON   VANES  271 

The  change  in  the  kinetic  energy  of  the  jet  is  equal  to  the  work  done  by  the  jet. 

U2 

The  kinetic  energy  per  Ib.  of  the  original  jet  is  —  and  the  final  kinetic  energy  is 

*9 

Hi! 

2<7  ' 

U2     U,2 
The  work  done  is,  therefore,  ^  --  -J-  ft.  Ibs.  and  the  efficien«y  is 


It  can  at  once  be  seen  from  the  geometry  of  the  figure  that 
Vv  =  U2  _  iy 

g  ~  20     20  ' 

For  AB2=AC2  +  CB2  +  2AC.CG, 

and  since  CD  =  CB  and  CD-'  =  AC2  +  AD2, 

therefore,  AB2  -  AD2  =  2  AC  (AC  +  CG) 

=  2uV. 

But  AB2-AD2  =  U2-U12, 

IP-T         vV 


.u      ,  - 

therefore, 

2< 

If  the  water  instead  of  leaving  the  vane  in  a  direction  perpendicular  to  v,  leaves 
it  with  a  velocity  Ul  having  a  component  Vl  parallel  to  v,  the  work  done  on  the 
vane  per  pound  of  water  is 


9 

If  Ul  be  drawn  on  the  figure  it  will  be  seen  that  the  change  of  velocity  in  the 

direction  of  motion  is  now  (V  -  VJ,  the  impressed  force  per  pound  is  —    —  1  ,  and 

/  V  —  V  \ 
the  work  done  is,  therefore,  (  —      l  \  vl  ft.  Ibs.  per  pound. 

As  before,  the  work  done  on  the  vane  is  the  loss  of  kinetic  energy  of  the  jet,  and 
therefore, 


9  20 

The  work  done  on  the  vane  per  pound  of  water  for  any  given  value  of  Uj  ,  is, 
therefore,  independent  of  the  direction  of  Uj  . 

Example  (2).  A  series  of  vanes  such  as  AB,  Fig.  175,  are  fixed  to  a  (turbine) 
wheel  which  revolves  about  a  fixed  centre  C,  with  an  angular  velocity  u. 

The  radius  of  B  is  K  and  of  A,  r.  Within  the  wheel  are  a  number  of  guide 
passages,  through  which  water  is  directed  with  a  velocity  U,  at  a  definite  inclination 
B  with  the  tangent  to  the  wheel.  The  air  is  supposed  to  have  free  access  to  the 
wheel. 

To  draw  the  triangles  of  velocity,  at  inlet  and  outlet,  and  to  find  the  directions 
of  the  tips  of  the  vanes,  so  that  the  water  moves  on  to  the  vanes  without  shock  and 
leaves  the  wheel  with  a  given  velocity  Uj.  Friction  neglected. 

As  in  the  last  example  the  velocity  relative  to  the  vane  must  remain  constant, 
and  therefore,  Vr  and  vr  are  equal,  but  v  and  vl  are  unequal. 

The  tangent  AH  to  the  vane  at  A  makes  an  angle  <£  with  the  tangent  AD  to  the 
wheel,  so  that  CD  makes  an  angle  <f>  with  AD.  The  triangle  of  velocities  ACD  at 
inlet  is,  therefore,  as  shown  in  the  figure  and  does  not  need  explanation. 

To  draw  the  triangle  of  velocities  at  exit,  set  out  BG  equal  to  vl  and  perpen- 
dicular to  the  radius  BC,  and  with  B  and  G  as  centres,  describe  circles  with  Ul  and 
vr  —  which  is  equal  to  Vr  —  as  radii  respectively,  intersecting  in  E.  Then  GE  is 
parallel  to  the  tangent  to  the  vane  at  B. 


272 


HYDRAULICS 


If  there  is  a  loss  of  head,  hf,  by  friction,  as  the  water  moves  over  the  vane  then 
vr  is  less  than  Vr,  if  hf  is  known,  it  can  be  found  from 


(See  Impulse  turbines.) 

Work  done  on  the  wheel.  Neglecting  friction  etc.  the  work  done  per  pound  of 
water  passing  through  the  wheel,  since  the  pressure  is  constant,  being  equal  to  the 
atmospheric  pressure,  is  the  loss  of  kinetic  energy  of  the  water,  and  is 

2 21"  ft*  ^S' 

The  work  done  on  the  wheel  can  also  be  found  from  the  consideration  of  the 
change  of  the  angular  momentum  of  the  water  passing  through  the  wheel.  Before 
going  on  however  to  determine  the  work  per  pound  by  this  method,  the  notation 
that  has  been  used  is  summarised  and  several  important  principles  considered. 


Notation  used  in  connection  with  vanes,  turbines  and  centrifugal 
pumps.  Let  U  be  the  velocity  with  which  the  water  approaches 
the  vane,  Fig.  175,  and  v  the  velocity,  perpendicular  to  the  radius 
AC,  of  the  edge  A  of  the  vane  at  which  water  enters  the  wheel. 

Let  Y  be  the  component  of  U  in  the  direction  of  v, 

u  the  component  of  U  perpendicular  to  v, 

Vr  the  relative  velocity  of  the  water  and  vane  at  A, 

Ui  the  velocity,  perpendicular  to  BC,  of  the  edge  B  of  the  vane 
at  which  water  leaves  the  wheel, 

Ui  the  velocity  with  which  the  water  leaves  the  wheel, 

Yi  the  component  of  Ui  in  the  direction  of  Vi, 


UNIVERSITY 

OF 


IMPACT   OF   WATER   ON   VANES 


273 


Ui  the  component  of  Ui  perpendicular  to  Vj,  or  along  BC, 
vr  the  relative  velocity  of  the  water  and  the  vane  at  B. 
Velocities  of  whirl.     The  component  velocities  V  and  Vi  are 

called  the  velocities   of  whirl   at   inlet   and  outlet  respectively. 

This  term  will  frequently  be  used  in  the  following  chapters. 

170.  Definition  of  angular  momentum. 

If  a  weight  of  W  pounds  is  moving  with  a  velocity  U,  Figs.  175 
and  176,  in  a  given  direction,  the  perpendicular  distance  of  which 
is  S  feet  from  a  fixed  centre  C,  the  angular  momentum  of  W  is 

W 

—  .  U  .  S  pounds  feet. 

171.  Change  of  angular  momentum. 

If  after  a  small  time  t  the  mass  is  moving  with  a  velocity  TJi  in 
a  direction,  which  is  at  a  perpendicular  distance  Si  from  C,  the 

W 

angular    momentum   is   now   —  UiSij    the    change   of    angular 

momentum  in  time  t  is 

W 

-(US-U.SO; 
y 

and  the  rate  of  change  of  angular  momentum  is 


Fig.  176.  Fig.  177. 

172.     Two  important  principles. 

(1)  Work  done  by  a  couple,  or  turning  moment.  When  a 
body  is  turned  through  an  angle  a  measured  in  radians,  under  the 
action  of  a  constant  turning  moment,  or  couple,  of  T  pounds  feet, 
the  work  done  is  Ta  foot  pounds. 

If  the  body  is  rotating  with  an  angular  velocity  o>  radians 
per  second,  the  rate  of  doing  work  is  T<o  foot  pounds  per  second, 

and  the  horse-power  is  ^=7:  . 


L.  II. 


18 


274  HYDRAULICS 

Suppose  a  body  rotates  about  a  fixed  centre  C,  Fig.  177,  and 
a  force  P  Ibs.  acts  on  the  body,  the  perpendicular  distance  from 
C  to  the  direction  of  P  being  S. 

The  moment  of  P  about  C  is 

T=P.S. 

If  the  body  turns  through  an  angle  w  in  one  second,  the 
distance  moved  through  by  the  force  P  is  w  .  S,  and  the  work 
done  by  P  in  foot  pounds  is 

P<oS=To>. 

And  since  one  horse-power  is  equivalent  to  33,000  foot  pounds 
per  minute  or  550  foot  pounds  per  second  the  horse-power  is 

HP-Ta> 
-550- 

(2)  The  rate  of  change  of  angular  momentum  of  a  body 
rotating  about  a  fixed  centre  is  equal  to  the  couple  acting  upon 
the  body.  Suppose  a  weight  of  W  pounds  is  moving  at  any  instant 
with  a  velocity  U,  Fig.  176,  the  perpendicular  distance  of  which 
from  a  fixed  centre  C  is  S,  and  that  a  couple  is  exerted  upon  W 
so  as  to  change  its  velocity  from  U  to  Ui  in  magnitude  and 
direction. 

The  reader  may  be  helped  by  assuming  the  velocity  U  is 
changed  to  Ui  by  a  wheel  such  as  that  shown  in  Fig.  175. 

Suppose  now  at  the  point  A  the  velocity  Ui  is  destroyed  in  a 
time  dt,  then  a  force  will  be  exerted  at  the  point  A  equal  to 

P  =  W   5 

~  g  'ct> 
and  the  moment  of  this  force  about  C  is  P  .  S. 

At  the  end  of  the  time  dt,  let  the  weight  W  leave  the  wheel 
with  a  velocity  Ui.  During  this  time  dt  the  velocity  Ui  might 
have  been  given  to  the  moving  body  by  a  force 


acting  at  the  radius  Si. 

The  moment  of  PI  is  PI  Si  ;  and  therefore  if  the  body  has  been 
acting  on  a  wheel,  Fig.  175,  the  reaction  of  the  wheel  thus  exerting 
the  couple  upon  the  body,  or  a  couple  has  been  exerted  upon  it  in 
any  other  way,  the  couple  required  to  change  the  velocity  of  W 
from  U  to  Ui  is 


a). 


Let  the  wheel  of  Fig.  175,  or  the  couple  which  is  acting  upon 
the  body,  have  an  angular  velocity  w. 


IMPACT  OF  WATER  ON  VANES  275 

In  a  time  dt  the  angle  moved  through  by  the  couple  is  wd£, 
and  therefore  the  work  done  in  time  dt  is 

W 

UiS,)  ..................  (2). 


Suppose  now  W  is  the  weight  of  water  in  pounds  per  second 
which  strikes  the  vanes  of  a  moving  wheel  of  any  form,  and  this 
water  has  its  velocity  changed  from  U  to  Ui,  then  by  making  dt 
in  either  equation  (1)  or  (2)  equal  to  unity,  the  work  done  per 
second  is 

W 


and  the  work  done  per  second  per  pound  of  water  entering  the 
wheel  is 

-(US-ILSO. 

I/ 

This  result,  as  will  be  seen  later  (page  337),  is  entirely  inde- 
pendent of  the  change  of  pressure  as  the  water  passes  through  the 
wheel,  or  of  the  direction  in  which  the  water  passes. 

173.  Work  done  on  a  series  of  vanes  fixed  to  a  wheel 
expressed  in  terms  of  the  velocities  of  whirl  of  the  water 
entering  and  leaving  the  wheel. 

Outward  flow  turbine.  If  water  enters  a  wheel  at  the  inner 
circumference,  as  in  Fig.  175,  the  flow  is  said  to  be  outward. 
On  reference  to  the  figure  it  is  seen  that  since  r  is  perpendicular 
to  Y,  and  S  to  U,  therefore 

r  _  U 

S~  V 

and  for  a  similar  reason 

R     Ui 

srvi- 

Again  the  angular  velocity  of  the  wheel 

V       Vi 

-=r  =  S' 

therefore  the  work  done  per  second  is 

W 

•-^•(y»-Vi«o, 

i/ 

and  the  work  done  per  pound  of  flow  is 


9         9 

Inward  flow  turbine.  If  the  water  enters  at  the  outer  cir- 
cumference of  a  wheel  with  a  velocity  of  whirl  V,  and  leaves  at 
the  inner  circumference  with  a  velocity  of  whirl  Vi,  the  velocities 

18—2 


276 


HYDRAULICS 


of  the  inlet  and  outlet  tips  of  the  vanes  being  v  and 
the  work  done  on  the  wheel  is  still 

V0 


respectively 


9          9 
The  flow  in  this  case  is  said  to  be  inward. 

Parallel  flow  or  axial  flow  turbine.  If  vanes,  such  as  those 
shown  in  Fig.  174,  are  fixed  to  a  wheel,  the  flow  is  parallel  to  the 
axis  of  the  wheel,  and  is  said  to  be  axial. 

For  any  given  radius  of  the  wheel,  Vi  is  equal  to  v,  and  the 
work  done  per  pound  is 


which  agrees  with  the  result  already  found  on  page  271. 

174.     Curved  vanes.    Pelton  wheel. 

Let  a  series  of  cups,  similar  to  Figs.  178  and  179,  be  moving 
with  a  velocity  v}  and  a  stream  with  a  greater  velocity  U  in  the 
same  direction. 

The  relative  velocity  is 

Neglecting  friction,  the  relative  velocity  Yr  will  remain  con- 
stant, and  the  water  will,  therefore,  leave  the  cup  at  the  point  B 
with  a  velocity,  Yr,  relative  to  the  cup. 


Fig.  178. 


If  the  tip  of  the  cup  at  B,  Fig.  178,  makes  an  angle  0  with  the 
direction  of  v,  the  absolute  velocity  with  which  the  water  leaves 
the  cup  will  be  the  vector  sum  of  v  and  Yr,  and  is  therefore  Ui  . 
The  work  done  on  the  cups  is  then 


IMPACT   OF   WATER   ON    VANES  277 

per  Ib.  of  water,  and  the  efficiency  is 


23 

For  Ui,  the  value 

Uj  -  J{v  -  (U  -  v)  cos  0}a  +  (U  -  v)*  sin  6»2 

can  be  substituted,  and  the  efficiency  thus  determined  in  terms  of 
v,  U  and  0. 

Pelton  wheel  cups.  If  0  is  zero,  as  in  Fig.  178,  and  U  —  v  is 
equal  to  v,  or  U  is  twice  v,  Ui  clearly  becomes  zero,  and  the  water 
drops  away  from  the  cup,  under  the  action  of  gravity,  without 
possessing  velocity  in  the  direction  of  motion. 

The  whole  of  the  kinetic  energy  of  the  jet  is  thus  absorbed 
and  the  theoretical  efficiency  of  the  cups  is  unity. 

The  work  done  determined  from  consideration  of  the  change  of 
momentum.  The  component  of  Ui,  Fig.  178,  in  the  direction  of 

motion,  is 

v  —  (U  -  v)  cos  0, 

and  the  change  of  momentum  per  pound  of  water  striking  the 
vanes  is,  therefore, 

U  -  v  +  (U  -  v)  cos  0 

9 

The  work  done  per  Ib.  is 


9 
and  the  efficiency  is 

-v+  (U  -v)  cos0} 


U2 
9  -29 


U2 
When  6  is  0,  cos  6  is  unity,  and 

_4a)(U-v) 
U 

which  is  a  maximum,  and  equal  to  unity,  when  v  is  ^-  . 

ft 

175.  Force  tending  to  move  a  vessel  from  which  water 
is  issuing  through  an  orifice. 

When  water  issues  from  a  vertical  orifice  of  area  a  sq.  feet, 
in  the  side  of  a  vessel  at  rest,  in  which  the  surface  of  the  water  is 
maintained  at  a  height  h  feet  above  the  centre  of  the  orifice,  the 


278  HYDRAULICS 

pressure  on  the  orifice,  or  the  force  tending  to  move  the  vessel 
in  the  opposite  direction  to  the  movement  of  the  water,  is 

¥  =  2w.a.h  Ibs., 
w  being  the  weight  of  a  cubic  foot  of  water  in  pounds. 

The  vessel  being  at  rest,  the  velocity  with  which  the  water 
leaves  the  orifice,  neglecting  friction,  is 


and  the  quantity  discharged  per  second  in  cubic  feet  is 

Q  =  av. 
The  momentum  given  to  the  water  per  second  is 

,  f     w  .a.v* 
M  =  — 

9 

=  2w  .a.h. 

But  the  momentum  given  to  the  water  per  second  is  equal  to 
the  impressed  force,  and  therefore  the  force  tending  to  move  the 
vessel  is 

Y**2w.a.h, 

or  is  equal  to  twice  the  pressure  that  would  be  exerted  upon  a 
plate  covering  the  orifice.  When  a  fireman  holds  the  nozzle  of  a 
hose-pipe  through  which  water  is  issuing  with  a  velocity  -u,  there 
is,  therefore,  a  pressure  on  his  hand  equal  to 

2wav2  _  wav2 

~W~     ~9~ 

If  the  vessel  has  a  velocity  V  backwards,  the  velocity  U  of  the 
water  relative  to  the  earth  is 

U=t>-V, 
and  the  pressure  exerted  upon  the  vessel  is 


\J 

The  work  done  per  second  is 


= 

g 

per  Ib.  of  flow  from  the  nozzle. 

The  efficiency  is  e  =  Y  (v  ~  Y) 

gfi 

=  2V(i7-Y) 

v2 
which  is  a  maximum,  when 

v  =  2V 
and  e  =  i. 


IMPACT   OF   WATER   ON   VANES  279 

176.     The  propulsion  of  ships  by  water  jets. 

A  method  of  propelling  ships  by  means  of  jets  of  water  issuing 
from  orifices  at  the  back  of  the  ship,  has  been  used  with  some 
success,  and  is  still  employed  to  a  very  limited  extent,  for  the 
propulsion  of  lifeboats. 

Water  is  taken  by  pumps  carried  by  the  ship  from  that 
surrounding  the  vessel,  and  is  forced  through  the  orifices.  Let 
v  be  the  velocity  of  the  water  issuing  from  the  orifice  relative 

to  the  ship,  and  Y   the   velocity   of  the   ship.     Then  ^-   is  the 

head  h  forcing  water  from  the  ship,  and  the  available  energy 
per  pound  of  water  leaving  the  ship  is  h  foot  pounds. 

The  whole  of  this  energy  need  not,  however,  be  given  to  the 
water  by  the  pumps. 

Imagine  the  ship  to  be  moving  through  the  water  and  having 
a  pipe  with  an  open  end  at  the  front  of  the  ship.  The  water  in 
front  of  the  ship  being  at  rest,  water  will  enter  the  pipe  with  a 

Y2 

velocity  Y  relative  to  the  ship,  and  having  a  kinetic  energy  -~- 

per  pound.  If  friction  and  other  losses  are  neglected,  the  work 
that  the  pumps  will  have  to  do  upon  each  pound  of  water  to  eject 
it  at  the  back  with  a  velocity  v  is,  clearly, 

<tf     Y2 


As  in  the  previous  example,  the  velocity  of  the  water  issuing 
from  the  nozzles  relative  to  the  water  behind  the  ship  is  v  -  Y, 

0—  V 

and  the  change  of  momentum  per  pound  is,  therefore,  —    -  .    If  & 

\y 

is  the  area  of  the  nozzles  the  propelling  force  on  the  ship  is 


and  the  work  done  is 


0 

The  efficiency  is  the  work  done  on  the  ship  divided  by  the 

work   done   by  the   engines,   which   equals   wav  (  ~-       -  )  and, 

\^9     ^9/ 
therefore, 

_2YO-Y) 

v2-Y2 

2Y 


280  HYDRAULICS 

which  can  be  made  as  near  unity  as  is  desired  by  making  v  and 
Y  approximate  to  equality. 

But  for  a  given  area  a  of  the  orifices,  and  velocity  v,  the  nearer 
v  approximates  to  Y  the  less  the  propelling  force  F  becomes,  and 
the  size  of  ship  that  can  be  driven  at  a  given  velocity  Y  for  the 
given  area  a  of  the  orifices  diminishes. 

Et?is2V,  e  =  f. 


EXAMPLES. 

(1)  Ten  cubic  feet  of  water  per  second  are  discharged  from  a  stationary 
jet,  the  sectional  area  of  which  is  1  square  foot.     The  water  impinges  nor- 
mally on  a  flat  surface,  moving  in  the  direction  of  the  jet  with  a  velocity 
of  2  feet  per  second.     Find  the  pressure  on  the  plane  in  Ibs.,  and  the  work 
done  on  the  plane  in  horse-power. 

(2)  A  jet  of  water  delivering  100  gallons  per  second  with  a  velocity  of 
20  feet  per  second  impinges  perpendicularly  on  a  wall.     Find  the  pressure 
on  the  wall. 

(3)  A  jet  delivers  160  cubic  feet  of  water  per  minute  at  a  velocity  of 
20  feet  per  second  and  strikes  a  plane  perpendicularly.     Find  the  pressure 
on  the  plane — (1)  when  it  is  at  rest ;  (2)  when  it  is  moving  at  5  feet  per 
second  in  the  direction  of  the  jet.     In  the  latter  case  find  the  work  done 
per  second  in  driving  the  plane. 

(4)  A  fire-engine  hose,  3  inches  bore,  discharges  water  at  a  velocity  of 
100  feet  per  second.     Supposing  the  jet  directed  normally  to  the  side  of  a 
building,  find  the  pressure. 

(5)  Water  issues  horizontally  from  a  fixed  thin-edged  orifice,  6  inches 
square,  under  a  head  of  25  feet.     The  jet  impinges  normally  on  a  plane 
moving  in  the  same  direction  at  10  feet  per  second.     Find  the  pressure  on 
the  plane  in  Ibs.,  and  the  work  done  in  horse-power.     Take  the  coefficient 
of  discharge  as  f64  and  the  coefficient  of  velocity  as  '97. 

(6)  A  jet  and  a  plane  surface  move  in  directions  inclined  at  30°,  with 
velocities  of  30  feet  and  10  feet  per  second  respectively.     What  is  the 
relative  velocity  of  the  jet  and  surface  ? 

(7)  Let  AB  and  BC  be  two  lines  inclined  at  30°.     A  jet  of  water  moves 
in  the  direction  AB,  with  a  velocity  of  25  feet  per  second,  and  a  series  of 
vanes  move  in  the  direction  CB  with  a  velocity  of  15  feet  per  second.    Find 
the  form  of  the  vane  so  that  the  water  may  come  on  to  it  tangentially,  and 
leave  it  in  the  direction  BD,  perpendicular  to  CB. 

Supposing  that  the  jet  is  1  foot  wide  and  1  inch  thick  before  impinging, 
find  the  effort  of  the  jet  on  the  vanes. 

(8)  A  curved  plate  is  mounted  on  a  slide  so  that  the  plate  is  free  to 
move  along  the  slide.     It  receives  a  jet  of  water  at  an  angle  of  30°  with  a 
normal  to  the  direction  of  sliding,  and  the  jet  leaves  the  plate  at  an  angle 


IMPACT   OF  WATER  ON   VANES  281 

of  120°  with  the  same  normal.  Find  the  force  which  must  be  applied  to 
the  plate  in  the  direction  of  sliding  to  hold  it  at  rest,  and  also  the  normal 
pressure  on  the  slide.  Quantity  of  water  flowing  is  500  Ibs.  per  minute 
with  a  velocity  of  35  feet  per  second. 

(9)  A  fixed  vane  receives  a  jet  of  water  at  an  angle  of  120°  with  a 
direction  AB.     Find  what  angle  the  jet  must  be  turned  through  in  order 
that  the  pressure  on  the  vane  in  the  direction  AB  may  be  40  Ibs.,  when  the 
flow  of  water  is  45  Ibs.  per  second  at  a  velocity  of  30  feet  per  second. 

(10)  Water  under  a  head  of  60  feet  is  discharged  through  a  pipe  6  inches 
diameter  and  150  feet  long,  and  then  through  a  nozzle,  the  area  of  which 
is  one-tenth  the  area  of  the  pipe. 

Neglecting  all  losses  but  the  friction  of  the  pipe,  determine  the  pressure 
on  a  fixed  plate  placed  in  front  of  the  nozzle. 

(11)  A  jet  of  water  4  inches  diameter  impinges  on  a  fixed  cone,  the 
axis  coinciding  with  that  of  the  jet,  and  the  apex  angle  being  30  degrees, 
at  a  velocity  of  10  feet  per  second.     Find  the  pressure  tending  to  move  the 
cone  in  the  direction  of  its  axis. 

(12)  A  vessel  containing  water  and  having  in  one  of  its  vertical  sides 
a   circular    orifice    1    inch    diameter,   which    at    first  is  plugged    up,    is 
suspended  in  such  a  way  that  any  displacing  force  can  be  accurately 
measured.     On  the  removal  of  the  plug,  the  horizontal  force  required  to 
keep  the  vessel  in  place,  applied  opposite  to  the  orifice,  is  3'6  Ibs.    By  the 
use  of  a  measuring  tank  the  discharge  is  found  to  be  31  gallons  per  minute, 
the  level  of  the  water  in  the  vessel  being  maintained  at  a  constant  height 
of  9  feet  above  the  orifice.     Determine  the  coefficients  of  velocity,  con- 
traction and  discharge. 

(13)  A  train  carrying  a  Ramsbottom's  scoop  for  taking  water  into  the 
tender  is  running  at  24  miles  an  hour.     What  is  the  greatest  height  at 
which  the  scoop  will  deliver  the  water  ?        1  £f  0^ 

(14)  A  locomotive  going  at  40  miles  an  hour  scoops  up  water  from  a 
trough.     The  tank  is  8  feet  above  the  mouth  of  the  scoop,  and  the  delivery 
pipe  has  an  area  of  50  square  inches.     If  half  the  available  head  is  wasted 
at  entrance,  find  the  velocity  at  which  the  water  is  delivered  into  the  tank, 
and  the  number  of  tons  lifted  in  a  trench  500  yards  long.     What,  under 
these  conditions,  is  the  increased  resistance;  and  what  is  the  minimum 
speed  of  train  at  which  the  tank  can  be  filled  ?     Lond.  Un.  1906. 

(15)  A   stream  delivering  3000  gallons  of  water  per  minute  with  a 
velocity  of  40  feet  per  second,  by  impinging  on  vanes  is  caused  freely  to 
deviate  through  an  angle  of  10°,  the  velocity  being  diminished  to  35  feet 
per  second.     Determine  the  pressure  on  the  vanes  due  to  impact.     If  the 
vanes  be  moving  in  the  direction  of  that  pressure,  find  their  velocity  and 
deduce  the  useful  horse-power.     Lond.  Un.  1906. 

(16)  Water  flows  from  a  2-inch  pipe,  without  contraction,  at  45  feet  per 
second. 

Determine  the  maximum  work  done  on  a  machine  carrying  moving 
plates  in  the  following  cases  and  the  respective  efficiencies : — 


282  HYDRAULICS 

(a)  When  the  water  impinges  on  a  single  flat  plate  at  right  angles  and 
leaves  tangentially. 

(b)  Similar  to  (a)  but  a  large  number  of  equidistant  flat  plates  are 
interposed  in  the  path  of  the  jet. 

(c)  When  the  water  glides  on  and  off  a  single  semi -cylindrical  cup. 

(d)  When  a  large  number  of  cups  are  used  as  in  a  Pelton  wheel. 

(17)  In  hydraulic  mining,  a  jet  6  inches  in  diameter,  discharged  under 
a  head  of  400  feet,  is  delivered  horizontally  against  a  vertical  cliff  face. 
Find  the  pressure  on  the  face.    What  is  the  horse-power  delivered  by  the 
jet? 

(18)  If  the  action  on  a  Pelton  wheel  is  equivalent  to  that  of  a  jet  on  a 
series  of  hemispherical  cups,  find  the  efficiency  when  the  speed  of  the 
wheel  is  five-eighths  of  the  speed  of  the  jet. 

(19)  If  in  the  last  question  the  jet  velocity  is  50  feet  per  second, 
and  the  jet  area  0'15  square  foot,  find  the  horse -power  of  the  wheel. 

(20)  A  ship  has  jet  orifices  3  square  feet  in  aggregate  area,  and  dis- 
charges through  the  jets  100  cubic  feet  of  water  per  second.     The  speed  of 
the  ship  is  15  feet  per  second.    Find  the  propelling  force  of  the  jets,  the 
efficiency  of  the  propeller,  and,  neglecting  friction,  the  horse-power  of  the 
engines. 


CHAPTER   IX. 

WATER  WHEELS   AND   TURBINES. 

Water  wheels  can  be  divided  into  two  classes  as  follows. 

(a)  Wheels  upon  which   the  water  does  work   partly  by 
impulse  but  almost  entirely  by  weight,  the  velocity  of  the  water 
when  it  strikes  the  wheel  being  small.     There  are  two  types  of 
this  class  of  wheel,  Overshot  Wheels,  Figs.  180   and  181,  and 
Breast  Wheels,  Figs.  182  and  184. 

(b)  Wheels  on  which  the  water  acts  by  impulse  as  when 
the  wheel  utilises  the  kinetic  energy  of  a  stream,  or  if  a  head  h  is 
available  the  whole  of  the  head  is  converted  into  velocity  before 
the  water  comes  in  contact  with  the  wheel.     In  most  impulse 
wheels  the  water  is  made  to  flow  under  the  wheel  and  hence 
they  are  called  Undershot  Wheels. 

It  will  be  seen  that  in  principle,  there  is  no  line  of  demarcation 
between  impulse  water  wheels  and  impulse  turbines,  the  latter 
only  differing  from  the  former  in  constructional  detail. 

177.     Overshot  water  wheels. 

This  type  of  wheel  is  not  suitable  for  very  low  or  very  high 
heads  as  the  diameter  of  the  wheel  cannot  be  made  greater  than 
the  head,  neither  can  it  conveniently  be  made  much  less. 

Figs.  180  and  181  show  two  arrangements  of  the  wheel,  the 
only  difference  in  the  two  cases  being  that  in  Fig.  181,  the  top  of 
the  wheel  is  some  distance  below  the  surface  of  the  water  in  the 
up-stream  channel  or  penstock,  so  that  the  velocity  v  with  which 
the  water  reaches  the  wheel  is  larger  than  in  Fig.  180.  This  has 
the  advantage  of  allowing  the  periphery  of  the  wheel  to  have  a 
higher  velocity,  and  the  size  and  weight  of  the  wheel  is  conse- 
quently diminished. 

The  buckets,  which  are  generally  of  the  form  shown  in  the 
figures,  or  are  curved  similar  to  those  of  Fig.  182,  are  con- 
nected to  a  rim  M  coupled  to  the  central  hub  of  the  wheel  by 


284 


HYDRAULICS 


suitable  spokes  or  framework.  This  class  of  wheel  has  been 
considerably  used  for  heads  varying  from  6  to  70  feet,  but  is  now 
becoming  obsolete,  being  replaced  by  the  modern  turbine,  which 
for  the  same  head  and  power  can  be  made  much  more  compact, 
and  can  be  run  at  a  much  greater  number  of  revolutions  per  unit 
time. 

E      «    D          K 


Fig.  180.     Overshot  Water  Wheel. 


Fig.  181.     Overshot  Water  Wheel. 

The  direction  of  the  tangent  to  the  blade  at  inlet  for  no  shock 
can  be  found  by  drawing  the  triangle  of  velocities  as  in  Figs.  180 
and  181.  The  velocity  of  the  periphery  of  the  wheel  is  v  and  the 
velocity  of  the  water  U.  The  tip  of  the  blade  should  be  parallel 
to  Vr.  The  mean  velocity  U,  of  the  water,  as  it  enters  the  wheel 


WATER   WHEELS  285 

in  Fig.  181,  will  be  v0  +  k  v/2^H,  v0  being  the  velocity  of  approach 
of  the  water  in  the  channel,  H  the  fall  of  the  free  surface  and  k 
a  coefficient  of  velocity.  The  water  is  generally  brought  to  the 
wheel  along  a  wooden  flume,  and  thus  the  velocity  U  and  the 
supply  to  the  wheel  can  be  maintained  fairly  constant  by  a  simple 
sluice  placed  in  the  flume. 

The  best  velocity  v  for  the  periphery  is,  as  shown  below, 
equal  to  JU  cos  0,  but  in  practice  the  velocity  v  is  frequently 
much  greater  than  this. 

In  order  that  U  may  be  about  2v  the  water  must  enter  the 
wheel  at  a  depth  not  less  than 

„    U2_2i;2 
=  ^=T 
below  the  water  in  the  penstock.     When 

v  =  4'5  feet,  H  =  0'63  feet, 
and  when  v  =  8  feet,      H  =  1  foot. 

If  the  total  fall  to  the  level  of  the  water  in  the  tail  race  is  h, 
the  diameter  of  the  wheel  may,  therefore,  be  between  h  and 

w 

ri . 

9 

Since  U  is  equal  to  j2gK,  for  given  values  of  U  and  of  h,  the 
larger  the  wheel  is  made  the  greater  must  be  the  angular  distance 
from  the  top  of  the  wheel  at  which  the  water  enters. 

With  the  type  of  wheel  and  penstock  shown  in  Fig.  181,  the 
head  H  is  likely  to  vary  and  the  velocity  U  will  not,  therefore,  be 
constant. 

If,  however,  the  wheel  is  designed  for  the  required  power  at 
minimum  flow,  when  the  head  increases,  and  there  is  a  greater 
quantity  of  water  available,  a  loss  in  efficiency  will  not  be 
important. 

The  horse-power  of  the  wheel.  Let  D  be  the  diameter  of  the 
wheel  in  feet  which  in  actual  wheels  is  from  10  to  70  feet. 

Let  N  be  the  number  of  buckets,  which  in  actual  wheels  is 
generally  from  2£  to  3D. 

Let  Q  be  the  volume  of  water  in  cubic  feet  of  water  supplied 
per  second. 

Let  w  be  the  angular  velocity  of  the  wheel  in  radians,  and  n 
the  number  of  revolutions  per  sec. 

Let  6  be  the  width  of  the  wheel. 

Let  d,  which  equals  r2  —  r} ,  be  the  depth  of  the  shroud,  which 
on  actual  wheels  is  from  10"  to  20". 


286  HYDRAULICS 


Whatever  the  form  of  the  buckets  the  capacity  of  each  bucket  is 

bd  .  -^-  ,  nearly. 
The  number  of  buckets  which  pass  the  stream  per  second  is 

N<*       AT 

~2^  =  ^'n' 

If  a  fraction  k  of  each  bucket  is  filled  with  water 


. 


or 


The  fraction  ~k  in  actual  wheels  is  from  J  to  ^  . 
If  h  is  the  fall  of  the  water  to  the  level  of  the  tail  race  and  e 
the  efficiency  of  the  wheel,  the  horse-power  is 

62-4  .  e  .  h 


550 

and  the  width  b  for  a  given  horse-power,  HP,  is 
1100HP          ?        HP 

^  />r\.  j      1     -iTHk      ~i  J.  /    w       -»     - 


Effect  of  centrifugal  forces.  As  the  wheel  revolves,  the  surface 
of  the  water  in  the  buckets,  due  to  centrifugal  forces,  takes  up  a 
parabolic  form. 

It  is  shown  on  page  335  that  when  a  mass  of  water  having  an 
inner  radius  n  and  outer  radius  r.2  revolves  about  a  fixed  centre 
with  angular  velocity  <«>,  the  pressure  head,  due  to  centrifugal 
forces,  at  any  radius  r,  is 

p  _  o>V  -  Q)Vi2 

w  2g 

To  balance  this  pressure  head  the  surface  of  the  water  in  any 
bucket,  at  the  point  C,  of  radius  r,  must  be  raised  above  the 
horizontal  through  A  a  distance 


This  is  the  equation  to  a  parabola,  and  the  surface  of  the  water, 
therefore,  assumes  the  form  of  a  parabolic  curve. 

Let  r0  be  the  radius  at  the  centre  of  the  surface  of  the  water  in 
any  cup  and  <f>  the  inclination  of  the  radius  r0  to  the  horizontal. 

Then  since  n  is  nearly  equal  to  rz  ,  TIO  r  =  r0  nearly. 


WATER  WHEELS  287 


Then  y  =  j-  (n  +  r)  (r-rj 

=  —  r0(r-ri)  nearly. 
y 

Therefore,  y  is  approximately  proportional  to  r-r1?  and  the 
surface  AB  is  approximately  a  straight  line  inclined  at  an  angle 
0,  the  tangent  of  which  is 

tan  0  =  —  -  cos  <£. 

Losses  of  energy  in  overshot  wheels. 

(a)     The  whole  of  the  velocity  head  -^-  is  lost  in  eddies  in  the 

buckets. 

In  addition,  as  the  water  falls  in  the  bucket  through  the 
vertical  distance  EM,  its  velocity  will  be  increased  by  gravity, 
and  the  velocity  thus  given  will  be  practically  all  lost  by  eddies. 

Again,  if  the  direction  of  the  tip  of  the  bucket  is  not  parallel  to 
Vr  the  water  will  enter  with  shock,  and  a  further  head  will  be 
lost.  The  total  loss  by  eddies  and  shock  may,  therefore,  be 
written 


or  7i!  +  &!£-, 

k  and  &i  being  coefficients  and  hi  the  vertical  distance  EM. 

(b)  The  water  begins  to  leave  the  buckets  before  the  level  of 
the  tail  race  is   reached.     This  is  increased  by  the  centrifugal 
forces,  as  clearly,  due  to  these  forces,  the  water  will  leave  the 
buckets  earlier  than  it  otherwise  would  do.     If  hm  is  the  mean 
height  above  the  tail  level  at  which  the  water  leaves  the  buckets, 
a  head  equal  to  hm  is  lost.     By  fitting  an  apron  GrH  in  front  of  the 
wheel  the  water  can  be  prevented  from  leaving  the  wheel  until  it 
is  very  near  the  tail  race. 

(c)  The  water  leaves  the  buckets  with  a  velocity  of  whirl 
equal  to  the  velocity  of  the  periphery  of  the  wheel  and  a  further 


2 

head    -  is  lost. 


(d)  If  the  level  of  the  tajl  water  rises  above  the  bottom  of 
the  wheel  there  will  be  a  further  loss  due  to,  (1)  the  head  h0  equal  to 
the  height  of  the  water  above  the  bottom  of  the  wheel,  (2)  the 
impact  of  the  tail  water  stream  on  the  buckets,  and  (3)  the 
tendency  for  the  buckets  to  lift  the  water  on  the  ascending  side  of 
the  wheel. 


288  HYDRAULICS 

In  times  of  flood  there  may  be  a  considerable  rise  of  the 
down-stream,  and  h0  may  then  be  a  large  fraction  of  In.  If  on 
the  other  hand  the  wheel  is  raised  to  such  a  height  above  the  tail 
water  that  the  bottom  of  the  wheel  may  be  always  clear,  the 
head  hm  will  be  considerable  during  dry  weather  flow,  and  the 
greatest  possible  amount  of  energy  will  not  be  obtained  from  the 
water,  just  when  it  is  desirable  that  no  energy  shall  be  wasted. 

If  h  is  the  difference  in  level  between  the  up  and  down-stream 
surfaces,  the  maximum  hydraulic  efficiency  possible  is 


h 

and  the  actual  hydraulic  efficiency  will  be 

/  y}2     v*\ 

h  -  (fefc,  +  Wo  +  hm  +  k^-  +  ^ 

e  = -TJ- 

&,  ki  and  A?0  being  coefficients. 

The  efficiency  as  calculated  from  equation  (1),  for  any  given 
value  of  hm,  is  a  maximum  when 


is  a  minimum. 

From  the  triangles  EKF  and  KDF,  Fig.  180, 

(U  cos  B  -  vY  +  (U  sin  0)2  -  Vr2. 
Therefore,  adding  v2  to  both  sides  of  the  equation, 
Yr2  +  v*  =  U2  cos2  B  -  2Uv  cos  6  +  Zv*  +  U2  sin2  0, 

which  is  a  minimum  for  a  given  value  of  U,  when  Wv  cos  0  —  W 
is  a  maximum.  Differentiating  and  equating  to  zero  this,  and 
therefore  the  efficiency,  is  seen  to  be  a  maximum,  when 

v  =  ~n  cos  0- 

The  actual  efficiencies  obtained  from  overshot  wheels  vary 
from  60  to  80  per  cent. 

178.    Breast  wheel. 

This  type  of  wheel,  like  the  overshot  wheel,  is  becoming 
obsolete.  Fig.  182  shows  the  form  of  the  wheel,  as  designed  by 
Fairbairn. 

The  water  is  admitted  to  the  wheel  through  a  number  of 
passages,  which  may  be  opened  or  closed  by  a  sluice  as  shown  in 
the  figure.  The  directions  of  these  passages  may  be  made  so  that 
the  water  enters  the  wheel  without  shock.  The  water  is  retained 


WATER   WHEELS 


289 


in  the  bucket,  by  the  breast,  until  the  bucket  reaches  the  tail  race, 
and  a  greater  fraction  of  the  head  is  therefore  utilised  than  in 
the  overshot  wheel.  In  order  that  the  air  may  enter  and  leave 
the  buckets  freely,  they  are  partly  open  at  the  inner  rim.  Since 
the  water  in  the  tail  race  runs  in  the  direction  of  the  motion  of 
the  bottom  of  the  wheel  there  is  no  serious  objection  to  the  tail 
race  level  being  6  inches  above  the  bottom  of  the  wheel. 

The  losses  of  head  will  be  the  same  as  for  the  overshot  wheel 
except  that  hm  will  be  practically  zero,  and  in  addition,  there  will 
be  loss  by  friction  in  the  guide  passages,  by  friction  of  the  water 
as  it  moves  over  the  breast,  and  further  loss  due  to  leakage 
between  the  breast  and  the  wheel. 


Fig.  182.    Breast  Wheel. 

According  to  Rankine  the  velocity  of  the  rim  for  overshot  and 
breast  wheels,  should  be  from  4J  to  8  feet  per  second,  and  the 
velocity  U  should  be  about  2v. 

The  depth  of  the  shroud  which  is  equal  to  r2  -  n  is  from  1  to 
If  feet.  Let  it  be  denoted  by  d.  Let  H  be  the  total  fall  and  let 
it  be  assumed  that  the  efficiency  of  the  wheel  is  65  per  cent.  Then, 
L.  H.  19 


290  HYDRAULICS 

the  quantity  of  water  required  per  second  in  cubic  feet  for  a 
given  horse-power  N  is 

0=        N.550 
^     62-4  x  H  x  0'65 
13-5N 

H 

From  |  to  §  of  the  volume  of  each  bucket,  or  from  \  to  §  of  the 
total  volume   of   the   buckets   on   the 
loaded  part  of  the  wheel  is  filled  with 
water. 

Let  b  be  the  breadth  of  the  buckets. 
If  now  v  is  the  velocity  of  the  rim,  and 
an  arc  AB,  Fig.  183,  is  set  off  on  the 
outer  rim  equal  to  v,  and  each  bucket 
is  half  full,  the  quantity  of  water 
carried  down  per  second  is 

JABCD.6. 

Therefore 


Q  =  if 


vdb. 


2r2 
Equating  this  value  of  Q  to  the  above  value,  the  width  b  is 

27ND 

"(n  +  rOwSH' 
D  being  the  outer  diameter  of  the  wheel. 

Breast  wheels  are  used  for  falls  of  from  5  to  15  feet  and  the 
diameter  should  be  from  12  to  25  feet.  The  width  may  be  as 
great  as  10  feet. 

Example.  A  breast  wheel  20  feet  diameter  and  6  feet  wide,  working  on  a  fall 
of  14  feet  and  having  a  depth  of  shroud  of  1'  3",  has  its  buckets  f  full.  The  mean 
velocity  of  the  buckets  is  5  feet  per  second.  Find  the  horse-power  of  the  wheel, 
assuming  the  efficiency  70  per  cent. 

TTT>     K     ,  0~      ,5     62-4x0-70x14' 

HP  =  5  x  l-2o  x  6  x  -  x -— 

o  550 

=  26-1. 

The  dimensions  of  this  wheel  should  be  compared  with  those  calculated  for  an 
inward  flow  turbine  working  under  the  same  head  and  developing  the  same  horse- 
power. See  page  339. 

179.     Sagebien  wheels. 

These  wheels,  Fig.  184,  have  straight  buckets  inclined  to  the 
radius  at  an  angle  of  from  30  to  45  degrees. 

The  velocity  of  the  periphery  of  the  wheel  is  very  small,  never 
exceeding  2|  to  3  feet  per  second,  so  that  the  loss  due  to  the  water 
leaving  the  wheel  with  this  velocity  and  due  to  leakage  between 
the  wheel  and  breast  is  small. 


WATER   WHEELS 


291 


An  efficiency  of  over  80  per  cent,  has  been  obtained  with 
these  wheels. 

The  water  enters  the  wheel  in  a  horizontal  direction  with 
a  velocity  U  equal  to  that  in  the  penstock,  and  the  triangle  of 
velocities  is  therefore  ABC. 

If  the  bucket  is  made  parallel  to  Vr  the  water  enters  without 
shock,  while  at  the  same  time  there  is  no  loss  of  head  due  to 
friction  of  guide  passages,  or  to  contraction  as  the  water  enters  or 
leaves  them ;  moreover  the  direction  of  the  stream  has  not  to  be 
changed. 


Fig.  184.     Sagebien  Wheel. 

The  inclined  straight  bucket  has  one  disadvantage ;  when  the 
lower  part  of  the  wheel  is  drowned,  the  buckets  as  they  ascend  are 
more  nearly  perpendicular  to  the  surface  of  the  tail  water  than 
when  the  blades  are  radial,  but  as  the  peripheral  speed  is  very 
low  the  resistance  due  to  this  cause  is  not  considerable. 

180.    Impulse  wheels. 

In  Overshot  and  Breast  wheels  the  work  is  done  principally 
by  the  weight  of  the  water.  In  the  wheels  now  to  be  considered 
the  whole  of  the  head  available  is  converted  into  velocity  before 
the  water  strikes  the  wheel,  and  the  work  is  done  on  the  wheel 
by  changing  the  momentum  of  the  mass  of  moving  water,  or  in 
other  words,  by  changing  the  kinetic  energy  of  the  water. 

19—2 


292 


HYDRAULICS 


Undershot  wheel  with  flat  blades.  The  simplest  case  is  when 
a  wheel  with  radial  blades,  similar  to  that  shown  in  Fig.  185,  is 
put  into  a  running  stream. 

If  6  is  the  width  of  the  wheel,  d  the  depth  of  the  stream  under 
the  wheel,  and  U  the  velocity  in  feet  per  second,  the  weight  of 
water  that  will  strike  the  wheel  per  second  is  b .  d .  w  .  U  Ibs.,  and 
the  energy  available  per  second  is 

U3 
b  .  d .  w  £-  foot  Ibs. 

Let  v.  be  the  mean  velocity  of  the  blades. 

The  radius  of  the  wheel  being  large  the  blades  are  similar  to 
a  series  of  flat  blades  moving  parallel  to  the  stream  and  the  water 
leaves  them  with  a  velocity  v  in  the  direction  of  motion. 

As  shown  on  page  268,  the  best  theoretical  value  for  the 
velocity  v  of  such  blades  is  JU  and  the  maximum  possible 
efficiency  of  the  wheel  is  0*5. 


Fig.  185.    Impulse  Wheel. 

By  placing  a  gate  across  the  channel  and  making  the  bed  near 
the  wheel  circular  as  in  Fig.  185,  and  the  width  of  the  wheel 
equal  to  that  of  the  channel,  the  supply  is  more  under  control,  and 
loss  by  leakage  is  reduced  to  a  minimum. 

The  conditions  are  now  somewhat  different  to  those  assumed 
for  the  large  number  of  flat  vanes,  and  the  maximum  possible 
efficiency  is  determined  as  follows. 

Let  Q  be  the  number  of  cubic  feet  of  water  passing  through 
the  wheel  per  second.  The  mean  velocity  with  which  the  water 
leaves  the  penstock  at  ab  is  U  =  fc  \/2gh.  Let  the  depth  of  the 


WATER  WHEELS  293 

stream  at  ab  be  t.  The  velocity  with  which  the  water  leaves  the 
wheel  at  the  section  cd  is  v,  the  velocity  of  the  blades.  If  the 
width  of  the  stream  at  cd  is  the  same  as  at  ab  and  the  depth 
is  h0,  then, 

h0  x  v  =  t  x  U, 

,      *U 
or  h0  =  —  . 

v 

Since  U  is  greater  than  v,  h0  is  greater  than  t,  as  shown  in 
the  figure. 

The  hydrostatic  pressure  on  the  section  cd  is  ?h0*bw  and  on 
the  section  ab  it  is  $Fbw. 

The  change  in  momentum  per  second  is 


and  this  must  be  equal  to  the  impressed  forces  acting  on  the  mass 
of  water  flowing  per  second  through  ab  or  cd. 

These  impressed  forces  are  P  the  driving  pressure  on  the  wheel 
blades,  and  the  difference  between  the  hydrostatic  pressures  acting 
on  cd  and  ab. 

If,  therefore,  the  driving  force  acting  on  the  wheel  is  P  Ibs., 
then, 

P  +  ±h<?bw  -  tfbw  =  ^  (U  -  v). 
Substituting  for  h0,  —  ,  the  work  done  per  second  is 


Or,  since  Q  =  b  .  t  .  U, 


The  efficiency  is  then, 

v  (U  -  v)  _  t_  /U  _  ^ 
g  2\v      I 


U 


which  is  a  maximum  when 

2t;2U2  -  4v3U  +  grrtJ*  +  gtv*  =  0. 

The  best  velocity,  v,  for  the  mean  velocity  of  the  blades,  has 
been  found  in  practice  to  be  about  0'4U,  the  actual  efficiency  is 
from  30  to  35  per  cent.,  and  the  diameters  of  the  wheel  are 
generally  from  10  to  23  feet. 

Floating  wheels.  To  adapt  the  wheel  to  the  rising  and 
lowering  of  the  waters  of  a  stream,  the  wheel  may  be  mounted  on 


294 


HYDRAULICS 


a  frame  which  may  be  raised  or  lowered  as  the  stream  rises,  or  the 
axle  carried  upon  pontoons  so  that  the  wheel  rises  automatically 
with  the  stream. 

181.     Poncelet  wheel. 

The  efficiency  of  the  straight  blade  impulse  wheels  is  very 
small,  due  to  the  large  amount  of  energy  lost  by  shock,  and  to  the 
velocity  with  which  the  water  leaves  the  wheel  in  the  direction  of 
motion. 

The  efficiency  of  the  wheel  is  doubled,  if  the  blades  are  of  such 
a  form,  that  the  direction  of  the  blade  at  entrance  is  parallel  to 
the  relative  velocity  of  the  water  and  the  blade,  as  first  suggested 
by  Poncelet,  and  the  water  is  made  to  leave  the  wheel  with  no 
component  in  the  direction  of  motion  of  the  periphery  of  the 
wheel. 

Fig.  186  shows  a  Poncelet  wheel. 


Fig.  186.     Undershot  Wheel. 

Suppose  the  water  to  approach  the  edge  A  of  a  blade  with  a 
velocity  U  making  an  angle  0  with  the  tangent  to  the  wheel  at  A. 

Then  if  the  direction  of  motion  of  the  water  is  in  the  direction 
AC,  the  triangle  of  velocities  for  entrance  is  ABC. 

The  relative  velocity  of  the  water  and  the  wheel  is  Yr,  and  if 
the  blade  is  made  sufficiently  deep  that  the  water  does  not  overflow 
the  upper  edge  and  there  is  no  loss  by  shock  and  by  friction,  a 
particle  of  water  will  rise  up  the  blade  a  vertical  height 


WATER  WHEELS  295 

It  then  begins  to  fall  and  arrives  at  the  tip  of  the  blade  with  the 
velocity  Vr  relative  to  the  blade  in  the  inverse  direction  BE. 

The  triangle  of  velocities  for  exit  is,  therefore,  ABE,  BE  being 
equal  to  BC. 

The  velocity  with  which  the  water  leaves  the  wheel  is  then 


It  has  been  assumed  that  no  energy  is  lost  by  friction  or  by 
shock,  and  therefore  the  work  done  on  the  wheel  is 


and  the  theoretical  hydraulic  efficiency*  is 

E- 

2. 

20 

.il.w  a) 

IP" 

This  will  be  a  maximum  when  Ui  is  a  minimum. 

Now  since  BE  =  BC,  the  perpendiculars  EF  and  CD,  on  to 
AB  and  AB  produced,  from  the  points  E  and  C  respectively,  are 
equal.  And  since  AC  and  the  angle  6  are  constant,  CD  is  constant 
for  all  values  of  v,  and  therefore  FE  is  constant.  But  AE,  that  is 
Ui,  is  always  greater  than  FE  except  when  AE  is  perpendicular 
to  AD.  The  velocity  Ui  will  have  its  minimum  value,  therefore, 
when  AE  is  equal  to  FE  or  Ui  is  perpendicular  to  v. 

The  triangles  of  velocities  are  then  as  in  Fig.  187,  the  point  B 
bisects  AD,  and 

For  maximum  efficiency,  therefore, 
v  =  ill  cos  6. 

*  In  what  follows,  the  terms  theoretical  hydraulic  efficiency  and  hydraulic 
efficiency  will  be  frequently  used.  The  maximum  work  per  Ib.  that  can  be  utilised 
by  any  hydraulic  machine  supplied  with  water  under  a  head  H,  and  from  which 

the  water  exhausts  with  a  velocity  u  is  H  - 1- .     The  ratio 

H-£ 


is  the  theoretical  hydraulic  efficiency.     If  there  are  other  hydraulic  losses  in  the 
machine  equivalent  to  a  head  hf  per  Ib.  of  flow,  the  hydraulic  efficiency  is 


The  actual  efficiency  of  the  machine  is  the  ratio  of  the  external  work  done  per  Ib. 
of  water  by  the  machine  to  H. 


296  HYDRAULICS 

The  efficiency  can  also  be  found  by  considering  the  change  of 
momentum. 

The  total  change  of  velocity  impressed  on  the  water  is  CE,  and 
the  change  in  the  direction  of  motion  is 
therefore  FD,  Fig.  186. 

And  since  BE  is  equal  to  BC,  FB  is 
equal  to  BD,  and  therefore, 

FD  =  2  (U  cos  0-v).  Fig.  187. 

The  work  done  per  Ib.  is,  then, 

2(Ucos0-?;) 


and  the  efficiency  is 


,, 


_  4  CUv  cos  6  -  v2)  ,0v 

~~ 


Differentiating  with  respect  to  v  and  equating  to  zero, 

Ucos0-20=0, 
or  v  =  ^U  cos  6. 

The  velocity  Ui  with  which  the  water  leaves  the  wheel,  is  then 
perpendicular  to  v  and  is 

Ui=Usm0. 

Substituting  for  v  its  value  JU  cos  6  in  (2),  the  maximum  efficiency 
is  cos2  6. 

The  same  result  is  obtained  from  equation  (1),  by  substituting 


The  maximum  efficiency  is  then 

,,     1     IP  sin2  0 

.L  =  1  --  Yya  —  =  cos  0. 

A  common  value  for  &  is  15  degrees,  and  the  theoretical 
hydraulic  efficiency  is  then  0*933. 

This  increases  as  0  diminishes,  and  would  become  unity  if  & 
could  be  made  zero. 

If,  however,  6  is  zero,  U  and  v  are  parallel  and  the  tip  of  the 
blade  will  be  perpendicular  to  the  radius  of  the  wheel. 

This  is  clearly  the  limiting  case,  which  practically  is  not 
realisable,  without  modifying  the  construction  of  the  wheel.  The 
necessary  modification  is  shown  in  the  Pelt-on  wheel  described  on 
page  377. 

The  actual  efficiency  of  Poncelet  wheels  is  from  55  to  65  per 
cent. 


WATER   WHEELS  297 

Form  of  the  bed.  Water  enters  the  wheel  at  all  points  between 
Q  and  R,  and  for  no  shock  the  bed  of  the  channel  PQ  should  be 
made  of  such  a  form  that  the  direction  of  the  stream,  where  it 
enters  the  wheel  at  any  point  A  between  R  and  Q,  should  make 
a  constant  angle  6  with  the  radius  of  the  wheel  at  A. 

With  0  as  centre,  draw  a  circle  touching  the  line  AS  which 
makes  the  given  angle  0  with  the  radius  AO.  Take  several 
other  points  on  the  circumference  of  the  wheel  between  R  and 
Q,  and  draw  tangents  to  the  circle  STV.  If  then  a  curve 
PQ  is  drawn  normal  to  these  several  tangents,  and  the  stream 
lines  are  parallel  to  PQ,  the  water  entering  any  part  of  the 
wheel  between  R  and  Q,  will  make  a  constant  angle  0  with  the 
radius,  and  if  it  enters  without  shock  at  A,  it  will  do  so  at  all 
points.  The  actual  velocity  of  the  water  U,  as  it  moves  along  the 
race  PQ,  will  be  less  than  \/2#H,  due  to  friction,  etc.  The 
coefficient  of  velocity  kv  in  most  cases  will  probably  be  between 
0'90  and  0'95,  so  that  taking  a  mean  value  for  kv  of  0*925, 

U  =  0-925 


The  best  value  for  the  velocity  v  taking  friction  into  account. 
In  determining  the  best  velocity  for  the  periphery  of  the  wheel  no 
allowance  has  been  made  for  the  loss  of  energy  due  to  friction  in 
the  wheel. 

If  Vr  is  the  relative  velocity  of  the  water  and  wheel  at  entrance, 
it  is  to  be  expected  that  the  velocity  relative  to  the  wheel  at  exit 
will  be  less  than  Vr,  due  to  friction  and  interference  of  the  rising 
and  falling  particles  of  water. 

The  case  is  somewhat  analogous  to  that  of  a  stone  thrown 
vertically  up  in  the  atmosphere  with  a  velocity  v.  If  there  were 
no  resistance  to  its  motion,  it  would  rise  to  a  certain  height, 


and  then  descend,  and  when  it  again  reached  the  earth  it  would 
have  a  velocity  equal  to  its  initial  velocity  v.  Due  to  resistances, 
the  height  to  which  it  rises  will  be  less  than  hi,  and  the  velocity 
with  which  it  reaches  the  ground  will  be  even  less  than  that  due 
to  falling  freely  through  this  diminished  height. 

Let  the  velocity  relative  to  the  wheel  at  exit  be  riVr,  n  being 
a  fraction  less  than  unity. 

The  triangle  of  velocities  at  exit  will  then  be  ABE,  Fig.  188. 
The  change  of  velocity  in  the  direction  of  motion  is  GH,  which 
equals 


298  HYDRAULICS 

If  the  velocity  at  exit  relative  to  the  wheel  is  only  nVr,  there 
must  have  been  lost  by  friction  etc.,  a  head  equal  to 


The  work  done  on  the  wheel  per  Ib.  of  water  is,  therefore, 

-v)}v     V,2 


9 


H 


Fig.  188. 

Let  (1  -  n.2)  be  denoted  by  /,  then  since 

Yr2  -  BH2  +  CH2  =  (U  cos  0  -  vy  +  U2  sin20, 
the  efficiency 


Differentiating  with  respect  to  v  and  equating  to  zero, 
2  (1  +  n)  U  cos  0  -  4  (1  +  w)  v  +  2U/  cos  6»  -  2t?/=  0, 
from  which 


= 


/+  2(1+71) 


If  /  is  now  supposed  to  be  0'5,  i.e.  the  head  lost  by  friction,  etc. 

.    0'5Vr2       .    Af71 

is  —  s  —  >  w  1S  v«l  an(i 

t;  =  '56U  cos  6. 
If  /is  taken  as  075, 

v  -  0'6U  cos  0. 

Dimensions  of  Poncelet  wheels.  The  diameter  of  the  wheel 
should  not  be  less  than  10  feet  when  the  bed  is  curved,  and  not 
less  than  15  feet  for  a  straight  bed,  otherwise  there  will  be  con- 
siderable loss  by  shock  at  entrance,  due  to  the  variation  of  the 
angle  6  which  the  stream  lines  make  with  the  blades  between  R 
and  Q,  Fig.  186.  The  water  will  rise  on  the  buckets  to  a  height 


WATER  WHEELS  299 

V  2 
nearly  equal  to  -£-  ,  and  since  the  water  first  enters  at  a  point  R, 

the  blade  depth  d  must,  therefore,  be  greater  than  this,  or  the 
water  will  overflow  at  the  upper  edge.  The  clearance  between 
the  bed  and  the  bottom  of  the  wheel  should  not  be  less  than  f  ". 
The  peripheral  distance  between  the  consecutive  blades  is  taken 
from  8  inches  to  18  inches. 

Horse-power  of  Poncelet  wheels.  If  H  is  the  height  of  the 
surface  of  water  in  the  penstock  above  the  bottom  of  the  wheel, 
the  velocity  U  will  be  about 

0' 
and  v  may  be  taken  as 


0'55  x  0'92  \2H  -  0'5 

Let  D  be  the  diameter  of  the  wheel,  and  b  the  breadth,  and  let 
t  be  the  depth  of  the  orifice  RP.  Then  the  number  of  revolutions 
per  minute  is 

0'5 


—  . 

7T.D 

The  coefficient  of  contraction  c  for  the  orifice  may  be  from  0*6, 
if  it  is  sharp-edged,  to  1  if  it  is  carefully  rounded,  and  may  be 
taken  as  0'8  if  the  orifice  is  formed  by  a  flat-edged  sluice. 

The  quantity  of  water  striking  the  wheel  per  second  is,  then, 


If  the  efficiency  is  taken  as  60  per  cent.,  the  work  done  per 
second  is  0'6  x  62'4QH  ft.  Ibs. 
The  horse-power  N  is  then 

34'5.c.£.  W2J7H.H 
550 

182.     Turbines. 

Although  the  water  wheel  has  been  developed  to  a  considerable 
degree  of  perfection,  efficiencies  of  over  80  per  cent,  having  been 
obtained,  it  is  being  almost  entirely  superseded  by  the  turbine. 

The  old  water  wheels  were  required  to  drive  slow  moving 
machinery,  and  the  great  disadvantage  attaching  to  them  of 
having  a  small  angular  velocity  was  not  felt.  Such  slow  moving 
wheels  are  however  entirely  unsuited  to  the  driving  of  modern 
machinery,  and  especially  for  the  driving  of  dynamos,  and  they 
are  further  quite  unsuited  for  the  high  heads  which  are  now 
utilised  for  the  generation  of  power. 

Turbine  wheels  on  the  other  hand  can  be  made  to  run  at  either 
low  or  very  high  speeds,  and  to  work  under  any  head  varying 


300  HYDRAULICS 

from  .  1  foot  to  2000  feet,  and  the  speed  can  be  regulated  with 
much  greater  precision. 

Due  to  the  slow  speeds,  the  old  water  wheels  could  not  develope 
large  power,  the  maximum  being  about  100  horse-power,  whereas 
at  Niagara  Falls,  turbines  of  10,000  horse-power  have  recently 
been  installed. 

Types  of  Turbines. 

Turbines  are  generally  divided  into  two  classes;  impulse,  or 
free  deviation  turbines,  and  reaction  or  pressure  turbines. 

In  both  kinds  of  turbines  an  attempt  is  made  to  shape  the 
vanes  so  that  the  water  enters  the  wheel  without  shock  ;  that  is 
the  direction  of  the  relative  velocity  of  the  water  and  the  vane  is 
parallel  to  the  tip  of  the  vane,  and  the  direction  of  the  leaving 
edge  of  the  vane  is  made  so  that  the  water  leaves  in  a  specified 
direction. 

In  the  first  class,  the  whole  of  the  available  head  is  converted 
into  velocity  before  the  water  strikes  the  turbine  wheel,  and  the 
pressure  in  the  driving  fluid  as  it  moves  over  the  vanes  remains 
constant,  and  equal  to  the  atmospheric  pressure.  The  wheel  and 
vanes,  therefore,  must  be  so  formed  that  the  air  has  free  access 
between  the  vanes,  and  the  space  between  two  consecutive  vanes 
must  not  be  full  of  water.  Work  is  done  upon  the  vanes,  or  in 
other  words,  upon  the  turbine  wheel  to  which  they  are  fixed,  in 
virtue  of  the  change  of  momentum  or  kinetic  energy  of  the 
moving  water,  as  in  examples  on  pages  270  —  2. 

Suppose  water  supplied  to  a  turbine,  as  in  Fig.  258,  under  an 
effective  head  H,  which  may  be  supposed  equal  to  the  total  head 
minus  losses  of  head  in  the  supply  pipe  and  at  the  nozzle.  The 
water  issues  from  the  nozzle  with  a  velocity  U  =  v/2grH,  and  the 
available  energy  per  pound  is 


Work  is  done  on  the  wheel  by  the  absorption  of  the  whole,  or 
part,  of  this  kinetic  energy. 

If  Ui  is  the  velocity  with  which  the  water  leaves  the  wheel, 
the  energy  lost  by  the  water  per  pound  is 

U2_UL2 
2<7      20' 

and  this  is  equal  to  the  work  done  on  the  wheel  together  with 
energy  lost  by  friction  etc..  in  the  wheel. 

In  the  second  class,  only  part  of  the  available  head  is  con- 
verted into  velocity  before  the  water  enters  the  wheel,  and  the 


TURBINES  301 

velocity  and  pressure  both  vary  as  the  water  passes  through  the 
wheel.  It  is  therefore  essential,  that  the  wheel  shall  always  be 
kept  full  of  water.  Work  is  done  upon  the  wheel,  as  will  be  seen 
in  the  sequence,  partly  by  changing  the  kinetic  energy  the  water 
possesses  when  it  enters  the  wheel,  and  partly  by  changing  its 
pressure  or  potential  energy. 

Suppose  water  is  supplied  to  the  turbine  of  Fig.  191,  under 
the  effective  head  H ;  the  velocity  U  with  which  the  water  enters 
the  wheel,  is  only  some  fraction  of  v/2#H,  and  the  pressure  head 
at  the  inlet  to  the  wheel  will  depend  upon  the  magnitude  of  U 
and  upon  the  position  of  the  wheel  relative  to  the  head  and  tail 
water  surfaces.  The  turbine  wheel  always  being  full  of  water, 
there  is  continuity  of  flow  through  the  wheel,  and  if  the  head 
impressed  upon  the  water  by  centrifugal  action  is  determined,  as 
on  page  335,  the  equations  of  Bemouilli  *  can  be  used  to  determine 
in  any  given  case  the  difference  of  pressure  head  at  the  inlet  and 
outlet  of  the  wheel. 

If  the  pressure  head  at  inlet  is  —  and  at  outlet  — ,  and  the 

velocity  with  which  the  water  leaves  the  wheel  is  Ui,  the  work 
done  on  the  wheel  (see  page  338)  is 

—  -  —  +  o 7p-  per  pound  of  water, 

w     w     2g      2g  ± 

or  work  is  done  on  the  wheel,  partly  by  changing  the  velocity 
head  and  partly  by  changing  the  pressure  head.  Such  a  turbine 
is  called  a  reaction  turbine,  and  the  amount  of  reaction  is  measured 
by  the  ratio 

p_pi 

w      w 

~TT~ 

Clearly,  if  p  is  made  equal  to  plf  the  limiting  case  is  reached, 
and  the  turbine  becomes  an  impulse,  or  free-deviation  turbine. 

It  should  be  clearly  understood  that  in  a  reaction  turbine  no 
work  is  done  on  the  wheel  merely  by  hydrostatic  pressure,  in  the 
sense  in  which  work  is  done  by  the  pressure  on  the  piston  of  a 
steam  engine  or  the  ram  of  a  hydraulic  lift. 

183.     Reaction  turbines. 

The  oldest  form  of  turbine  is  the  simple  reaction,  or  Scotch 
turbine,  which  in  its  simplest  form  is  illustrated  in  Fig.  189. 

A  vertical  tube  T  has  two  horizontal  tubes  connected  to  it,  the 
outer  ends  of  which  are  bent  round  at  right  angles  to  the  direction 

*  See  page  334. 


302 


HYDKAULICS 


of  length  of  the  tube,  or  two  holes  0  and  Oi  are  drilled  as  in  the 
figure. 

Water  is  supplied  to  the  central  tube  at  such  a  rate  as  to  keep 
the  level  of  the  water  in  the  tube 
constant,  and  at  a  height  h  above 
the  horizontal  tubes.  Water  escapes 
through  the  orifices  0  and  Oi  and 
the  wheel  rotates  in  a  direction 
opposite  to  the  direction  of  flow  of 
the  water  from  the  orifices.  Tur- 
bines of  this  class  are  frequently 
used  to  act  as  sprinklers  for  distri- 
buting  liquids,  as  for  example  for 
distributing  sewage  on  to  bacteria  Fig.  189.  Scotch  Turbine, 
beds. 

A  better  practical  form,  known  as  the  Whitelaw  turbine,  is 
shown  in  Fig.  190. 


i 


Fig.  190.    Whitelaw  Turbine. 

To  understand  the  action  of  the  turbine  it  is  first  necessary  to 
consider  the  effect  of  the  whirling  of  the  water  in  the  arm  upon 


TURBINES  303 

the  discharge  from  the  wheel.     Let  v  be  the  velocity  of  rotation 
of  the  orifices,  and  h  the  head  of  water  above  the  orifices. 

Imagine  the  wheel  to  be  held  at  rest  and  the  orifices  opened  ; 
then  the  head  causing  velocity  of  flow  relative  to  the  arm  is 
simply  7i,  and  neglecting  friction  the  water  will  leave  the  nozzle 
with  a  velocity 

v0  =  \f2gh. 

Now  suppose  the  wheel  is  filled  with  water  and  made  to  rotate 
at  an  angular  velocity  w,  the  orifices  being  closed.  There  will 
now  be  a  pressure  head  at  the  orifice  equal  to  h  plus  the  head 
impressed  on  the  water  due  to  the  whirling  of  each  particle  of 
water  in  the  arm. 

Assume  the  arm  to  be  a  straight  tube,  Fig.  189,  having  a  cross 
sectional  area  a.  At  any  radius  r  take  an  element  of  thickness  dr. 

The  centrifugal  force  due  to  this  element  is 

-,  ,.    w  .  a  .  wVdr 
of  =  --       -  . 
9 

The  pressure  per  unit  area  at  the  outer  periphery  is,  therefore, 

R 


a  Jo 


and  the  head  impressed  on  the  water  is 

p       eoV2 

w~  2g  ' 
Let  v  be  the  velocity  of  the  orifice,  then  v  =  wr,  and  therefore 

p  _  ir 
w~2g' 

If  now  the  wheel  be  assumed  frictionless  and  the  orifices  are 
opened,  and  the  wheel  rotates  with  the  angular  velocity  w,  the 
head  causing  velocity  of  flow  relative  to  the  wheel  is 

H_  =  h  +  2-=h+£  ...(1). 

w  2g 

Let  Vr  be  the  velocity  relative  to  the  wheel  with  which  the 
water  leaves  the  orifice. 


The  velocity  relative  to  the  ground,  with  which  the  water 
leaves  the  wheel,  is  Vr  —  v,  the  vector  sum  of  Vr  and  v. 


304  HYDRAULICS 

The  water  leaves  the  wheel,  therefore,  with  a  velocity  relative 
to  the  ground  of  p  =  Vr  -  v,  and  the  kinetic  energy  lost  is 

— ^ per  pound  of  water. 

The  theoretical  hydraulic  efficiency  is  then, 


_  2V  (Vr  ~  V} 

Vr2-t?2 
2v 

=  Vr  +  V* 

Since  from  (2),  Vr  becomes  more  nearly  equal  to  v  as  v 
increases,  the  energy  lost  per  pound  diminishes  as  v  increases, 
and  the  efficiency  E,  therefore,  increases  with  v. 

The  efficiency  of  the  reaction  wheel  when  friction  is  considered. 
As  before, 


JcV  2 
Assuming  the  head  lost  by  friction  to  be  —^- ,  the  total  head 

must  be  equal  to 

H=^  +  |-=  — ^-  (1  +  fc) (4). 

The  work  done  on  the  wheel,  per  pound,  is  now 

Z.~U"  2  .,2 


and  the  hydraulic  efficiency  is 

1,~\T  2 


Substituting  for  h  from  (4)  and  for  ft,  Vr  -  v, 

2v(Vr-v) 
(l+k)Vr*-V2' 

Let  Vr  =  nv, 

then  ^2(- 


Differentiating  and  equating  to  zero, 


From  which 


TURBINES 

Tfc 


305 


+  k' 


Or  the  efficiency  is  a  maximum  when 


and 


v. 


Fig.  191.     Outward  Flow  Turbine. 


L.  H. 


306  HYDRAULICS 

184.     Outward  flow  turbines. 

The  outward  flow  turbine  was  invented  in  1828  by  Four- 
neyron.  A  cylindrical  wheel  W,  Figs.  191,  192,  and  201,  having 
a  number  of  suitably  shaped  vanes,  is  fixed  to  a  vertical  axis. 
The  water  enters  a  cylindrical  chamber  at  the  centre  of  the 
turbine,  and  is  directed  to  the  wheel  by  suitable  fixed  guide 
blades  Gr,  and  flows  through  the  wheel  in  a  radial  direction 
outwards.  Between  the  guide  blades  and  the  wheel  is  a  cylindri- 
cal sluice  E  which  is  used  to  control  the  flow  of  water  through 
the  wheel. 


Fig.  191  a. 

This  method  of  regulating  the  flow  is  very  imperfect,  as  when 
the  gate  partially  closes  the  passages,  there  must  be  a  sudden 
enlargement  as  the  water  enters  the  wheel,  and  a  loss  of  head 
ensues.  The  efficiency  at  "part  gate"  is  consequently  very 
much  less  than  when  the  flow  is  unchecked.  This  difficulty  is 
partly  overcome  by  dividing  the  wheel  into  several  distinct 
compartments  by  horizontal  diaphragms,  as  shown  in  Fig.  192, 
so  that  when  working  at  part  load,  only  the  efficiency  of  one 
compartment  is  affected. 

The  wheels  of  outward  flow  turbines  may  have  their  axes, 
either  horizontal  or  vertical,  and  may  be  put  either  above,  or 
below,  the  tail  water  level. 

The  "suction  tube"  If  placed  above  the  tail  water,  the 
exhaust  must  take  place  down  a  "suction  pipe,"  as  in  Fig.  201, 
page  317,  the  end  of  which  must  be  kept  drowned,  and  the  pipe 
air-tight,  so  that  at  the  outlet  of  the  wheel  a  pressure  less  than 
the  atmospheric  pressure  may  be  maintained.  If  hi  is  the  height 
of  the  centre  of  the  discharge  periphery  of  the  wheel  above  the 
tail  water  level,  and  pa  is  the  atmospheric  pressure  in  pounds  per 
square  foot,  the  pressure  head  at  the  discharge  circumference  is 

^-^  =  34-^. 
w 


TURBINES 


307 


The  wheel  cannot  be  more  than  34  feet  above  the  level  of  the  tail 
water,  or  the  pressure  at  the  outlet  of  the  wheel  will  be  negative, 
and  practically,  it  cannot  be  greater  than  25  feet. 

It  is  shown  later  that  the  effective  head,  under  which  the 
turbine  works,  whether  it  is  drowned,  or  placed  in  a  suction  tube, 
is  H,  the  total  fall  of  the  water  to  the  level  of  the  tail  race. 


Fig.  192.     Fourneyron  Outward  Flow  Turbine. 

The  use  of  the  suction  tube  has  the  advantage  of  allowing  the 
turbine  wheel  to  be  placed  at  some  distance  above  the  tail  water 
level,  so  that  the  bearings  can  be  readily  got  at,  and  repairs  can 
be  more  easily  executed. 

By  making  the  suction  tube  to  enlarge  as  it  descends,  the 
velocity  of  exit  can  be  diminished  very  gradually,  and  its  final 

20—2 


308  HYDRAULICS 

value  kept  small.  If  the  exhaust  takes  place  direct  from  the 
wheel,  as  in  Fig.  192,  into  the  air,  the  mean  head  available  is  the 
head  of  water  above  the  centre  of  the  wheel. 

Triangles  of  velocities  at  inlet  and  outlet.  For  the  water  to 
enter  the  wheel  without  shock,  the  relative  velocity  of  the  water 
and  the  wheel  at  inlet  must  be  parallel  to  the  inner  tips  of  the 
vanes.  The  triangles  of  velocities  at  inlet  and  outlet  are  shown 
in  Figs.  193  and  194. 


Fig.  194. 

Let  AC,  Fig.  193,  be  the  velocity  U  in  direction  and  magnitude 
of  the  water  as  it  flows  out  of  the  guide  passages,  and  let  AD  be 
the  velocity  v  of  the  receiving  edge  of  the  wheel.  Then  DC  is  Yr 
the  relative  velocity  of  the  water  and  vane,  and  the  receiving 
edge  of  the  vane  must  be  parallel  to  DC.  The  radial  component 
GrC,  of  AC,  determines  the  quantity  of  water  entering  the  wheel 
per  unit  area  of  the  inlet  circumference.  Let  this  radial  velocity 
be  denoted  by  u.  Then  if  A  is  the  peripheral  area  of  the  inlet 
face  of  the  wheel,  the  number  of  cubic  feet  Q  per  second  entering 
the  wheel  is 

Q=A.th 

or,  if  d  is  the  diameter  and  b  the  depth  of  the  wheel  at  inlet,  and 
t  is  the  thickness  of  the  vanes,  and  n  the  number  of  vanes, 

Q  =  (ird  -  n .  t)  .  b  .  u. 

Let  D  be  the  diameter,  and  AI  the  area  of  the  discharge  peri- 
phery of  the  wheel. 

The  peripheral  velocity  Vi  at  the  outlet  circumference  is 


TURBINES  309 

Let  HI  be  the  radial  component  of  velocity  of  exit,  then  what- 
ever the  direction  with  which  the  water  leaves  the  wheel  the 
radial  component  of  velocity  for  a  given  discharge  is  constant. 

The  triangle  of  velocity  can  now  be  drawn  as  follows  : 

Set  off  BE  equal  to  Vi,  Fig.  194,  and  BK  radial  and  equal 
to  UL 

Let  it  now  be  supposed  that  the  direction  EF  of  the  tip  of  the 
vane  at  discharge  is  known.  Draw  EF  parallel  to  the  tip  of  the 
vane  at  D,  and  through  K  draw  KF  parallel  to  BE  to  meet  EF 
in  F. 

Then  BF  is  the  velocity  in  direction  and  magnitude  with  which 
the  water  leaves  the  wheel,  relative  to  the  ground,  or  to  the  fixed 
casing  of  the  turbine.  Let  this  velocity  be  denoted  by  Ui.  If, 
instead  of  the  direction  EF  being  given,  the  velocity  Ui  is  given 
in  direction  and  magnitude,  the  triangle  of  velocity  at  exit  can  be 
drawn  by  setting  out  BE  and  BF  equal  to  vl  and  Ui  respectively, 
and  joining  EF.  Then  the  tip  of  the  blade  must  be  made  parallel 
toEF. 

For  any  given  value  of  Ui  the  quantity  of  water  flowing 
through  the  wheel  is 


Work  done  on  the  wheel  neglecting  friction,  etc.     The  kinetic 
energy  of  the  water  as  it  leaves  the  turbine  wheel  is 

Uj2 

2~  per  pound, 

and  if  the  discharge  is  into  the  air  or  into  the  tail  water  this 
energy  is  of  necessity  lost.  Neglecting  friction  and  other  losses, 
the  available  energy  per  pound  of  water  is  then 

H-^footlbs., 
and  the  theoretical  hydraulic  efficiency  is 


and  is  constant  for  any  given  value  of  Ui,  and  independent  of  the 
direction  of  Ui.  This  efficiency  must  not  be  confused  with  the 
actual  efficiency,  which  is  much  less  than  E. 

The  smaller  Ui,  the  -greater  the  theoretical  hydraulic  efficiency, 
and  since  for  a  given  flow  through  the  wheel,  Ui  will  be  least 
when  it  is  radial  and  equal  to  Ui,  the  greatest  amount  of  work 
will  be  obtained  for  the  given  flow,  or  the  efficiency  will  be  a 
maximum,  when  the  water  leaves  the  wheel  radially.  If  the 


310  HYDRAULICS 

water  leaves  with  a  velocity  Ui  in  any  other  direction,  the 
efficiency  will  be  the  same,  but  the  power  of  the  wheel  will  be 
diminished.  If  the  discharge  takes  place  down  a  suction  tube, 
and  there  is  no  loss  between  the  wheel  and  the  outlet  from  the 
tube,  the  velocity  head  lost  then  depends  upon  the  velocity  Ui 
with  which  the  water  leaves  the  tube,  and  is  independent  of  the 
velocity  or  direction  with  which  the  water  leaves  the  wheel. 

The  velocity  of  whirl  at  inlet  and  outlet.  The  component  of 
U,  Fig.  193,  in  the  direction  of  v  is  the  velocity  of  whirl  at  inlet, 
and  the  component  of  Ui,  Fig.  194,  in  the  direction  of  t?i,  is  the 
velocity  of  whirl  at  exit. 

Let  Y  and  Vi  be  the  velocities  of  whirl  at  inlet  and  outlet 
respectively,  then 


and  Vi  =  Ui  sin  ft  =  Ui  tan  /?. 

Work  done  on  the  wheel.  It  has  already  been  shown, 
section  173,  page  275,  that  when  water  enters  a  wheel,  rotating 
about  a  fixed  centre,  with  a  velocity  U,  and  leaves  it  with  velocity 
Ui,  the  component  Vi  of  which  is  in  the  same  direction  as  viy  the 
work  done  on  the  wheel  is 


Vv        !! 

-  per  pound, 
9          9 

and  therefore,  neglecting  friction, 

5p.5a.H-g  .....................  (i). 

99  %9 

This  is  a  general  formula  for  all  classes  of  turbines  and  should 
be  carefully  considered  by  the  student. 
Expressed  trigonometrically, 

Vi  HI  tan  /3  _  ^     Ui2  ,0. 

- 


...............        < 

9  9  %9 

If  F  is  to  the  left  of  BK,  V»  is  negative. 

Again,  since  the  radial  flow  at  inlet  must  equal  the  radial  flow 
at  outlet,  therefore 

AU8in0  =  AiUiCosj8   .....................  (3). 

When  Ui  is  radial,  YI  is  zero,  and  u\  equals  v\  tan  a. 


£  i-i  TT       i  /r, 

from  which  -  ^H--2-^  -  .....................  (5), 

y  ^y 

and  from  (3)  AU  sin  0  =  A^  tan  a   .  .  .  .  (6)  . 


TURBINES  311 

If  the  tip  of  the  vane  is  radial  at  inlet,  i.e.  Yr  is  radial, 


and 


tan2  a, 


2(7 


.(7) 
(8). 


„  . 

In  actual  turbines     -  is  from  *02H  to  '07H. 


Example.  An  outward  flow  turbine  wheel,  Fig.  195,  has  an  internal  diameter  of 
5*249  feet,  and  an  external  diameter  of  6-25  feet,  and  it  makes  250  revolutions  per 
minute.  The  wheel  has  32  vanes,  which  may  be  taken  as  f  inch  thick  at  inlet  and 
1£  inches  thick  at  outlet.  The  head  is  141-5  feet  above  the  centre  of  the  wheel  and 
the  exhaust  takes  place  into  the  atmosphere.  The  effective  width  of  the  wheel  face 
at  inlet  and  outlet  is  10  inches.  The  quantity  of  water  supplied  per  second  is 
215  cubic  feet. 

Neglecting  all  frictional  losses,  determine  the  angles  of  the  tips  of  the  vanes  at 
inlet  and  outlet  so  that  the  water  shall  leave  radially. 

The  peripheral  velocity  at  inlet  is 

v  =  ir  x  5-249  x  -Vu—69  ft.  per  sec., 
and  at  outlet  ^  =  w  x  6-25  x  a#p  =  82  ft.      „     „ 


Fig.  195. 

The  radial  velocity  of  flow  at  inlet  is 

215 


~ir  x  5-249  x^-f 
=  18-  35  ft.  per  sec. 
The  radial  velocity  of  flow  at  exit  is 

215 


Therefore, 


=  16-5  ft.  per  sec. 
^-"  =  4-23  ft. 


312 


HYDRAULICS 


Then 


—  =  141-5-4-23 


and 


V= 


:  137-27  ft. 
137-27  x  32-2 
69 


=  64  ft.  per  sec. 


To  draw  the  triangle  of  velocities  at  inlet  set  out  v  and  u  at  right  angles. 

Then  since  V  is  64,  and  is  the  tangential  component  of  U,  and  u  is  the  radial 
component  of  U,  the  direction  and  magnitude  of  U  is  determined. 

By  joining  B  and  C  the  relative  velocity  Vr  is  obtained,  and  BC  is  parallel  to  the 
tip  of  the  vane. 

The  triangle  of  velocities  at  exit  is  DEF,  and  the  tip  of  the  vane  must  be  parallel 
toEF. 


Fig.  196. 


Fig.  197. 


—  Outlet— 


The  angles  0,  <f>,  and  a  can  be  calculated ;  for 


and 

and,  therefore, 


6-4 
18-35 


=  -3-670 


tan  a= 


=  0-1994, 


0*16°, 

0  =  105°  14', 
a  =  11°  17'. 


It  will  be  seen  later  how  these  angles  are  modified  when  friction  is  considered. 

Fig.  198  shows  the  form  the  guide  blades  and  vanes  of  the  wheel  would 
probably  take. 

The  path  of  the  water  through  the  wheel.  The  average  radial  velocity  through 
the  wheel  may  be  taken  as  17-35  feet. 

The  time  taken  for  a  particle  of  water  to  get  through  the  wheel  is,  therefore, 


The  angle  turned  through  by  the  wheel  in  this  time  is  0-39  radians. 

Set  off  the  arc  AB,  Fig.  198,  equal  to  -39  radian,  and  divide  it  into  four  equal 
parts,  and  draw  the  radii  eatfb,  gc  and  Bd. 

Divide  AD  also  into  four  equal  parts,  and  draw  circles  through  Ax,  A2,  and  A3. 

Suppose  a  particle  of  water  to  enter  the  wheel  at  A  in  contact  with  a  vane  and 
suppose  it  to  remain  in  contact  with  the  vane  during  its  passage  through  the  wheel. 
Then,  assuming  the  radial  velocity  is  constant,  while  the  wheel  turns  through  the 
arc  Ae  the  water  will  move  radially  a  distance  AAj  and  a  particle  that  came  on  to 


TURBINES 


313 


the  vane  at  A  will,  therefore,  be  in  contact  with  the  vane  on  the  arc  through  Ax . 
The  vane  initially  passing  through  A  will  be  now  in  the  position  el,  al  being 
equal  to  hJ  and  the  particle  will  therefore  be  at  1.  When  the  particle  arrives  on 
the  arc  through  A2  the  vane  will  pass  through  /,  and  the  particle  will  consequently 
be  at  2,  62  being  equal  to  mn.  The  curve  A4  drawn  through  Al  2  etc.  gives  the 
path  of  the  water  relative  to  the  fixed  casing. 


Fig.  198. 

185.  Losses  of  head  due  to  frictional  and  other  resistances 
in  outward  flow  turbines. 

The  losses  of  head  may  be  enumerated  as  follows  : 

(a)  Loss  by  friction  at  the  sluice  and  in  the  penstock  or 
supply  pipe. 

If  v0  is  the  velocity,  and  ha  the  head  lost  by  friction  in 
the  pipe, 


(6)  As  the  water  enters  and  moves  through  the  guide 
passages  there  will  be  a  loss  due  to  friction  and  by  sudden  changes 
in  the  velocity  of  flow. 

This  head  may  be  expressed  as 


Jc  being  a  coefficient. 


*  See  page  119. 


314 


HYDRAULICS 


(c)  There  is  a  loss  of  head  at  entrance   due  to  shock  as 
the   direction   of    the   vane   at  entrance   cannot   be   determined 
with  precision. 

This  may  be  written 

Vr2 
Iu(*  —  n/i  "7^       * 

2# 

that  is,  it  is  made  to  depend  upon  Vr  the  relative  velocity  of  the 
water,  and  the  tip  of  the  vane. 

(d)  In  the  wheel  there  is  a  loss  of  head  hd,  due  to  friction, 
which  depends  upon  the  relative  velocity  of   the  water  and  the 
wheel.     This  relative  velocity  may  be  changing,  and  on  any  small 
element  of  surface  of  the  wheel  the  head  lost  will  diminish,  as  the 
relative  velocity  diminishes. 

It  will  be  seen  on  reference  to  Figs.  193  and  194,  that  as  the 
velocity  of  whirl  Vi  is  diminished  the  relative  velocity  of  flow  vr  at 
exit  increases,  but  the  relative  velocity  Yr  at  inlet  passes  through 
a  minimum  when  V  is  equal  to  v,  or  the  tip  of  the  vane  is  radial. 
If  V0  is  the  relative  velocity  of  the  water  and  the  vane  at  any 
radius,  and  b  is  the  width  of  the  vane,  and  dl  an  element  of 
length,  then, 


&2  being  a  third  coefficient. 

If  there  is  any  sudden  change  of  velocity  as  the  water  passes 
through  the  wheel  there  will  be  a  further  loss,  and  if  the  turbine 
has  a  suction  tube  there  may  be  also  a  small  loss  as  the  water 
enters  the  tube  from  the  wheel. 

The  whole  loss  of  head  in  the  penstock  and  guide  passages  may 
be  called  H/  and  the  loss  in  the  wheel  hf.  Then  if  U0  is  the 


Rotor 


Boyden  Dtffuser 


Fiaed 


Fig.  199. 


TURBINES  315 

velocity  with  which  the  water  leaves  the  turbine  the  effective 
head  is 


In  well  designed  inward  and  outward  flow  turbines 


varies  from  O'lOH  to  *22H  and  the  hydraulic  efficiency  is,  therefore, 
from  90  to  78  per  cent. 

The  efficiency  of  inward  and  outward  flow  turbines  including 
mechanical  losses  is  from  75  to  88  per  cent. 

Calling  the  hydraulic  efficiency  e}  the  general  formula  (1), 
section  184,  may  now  be  written 


9         9 

=  '78to'9H. 

Outward  flow  turbines  were  made  by  Boy  den*  about  1848  for 
which  he  claimed  an  efficiency  of  88  per  cent.  The  workmanship 
was  of  the  highest  quality  and  great  care  was  taken  to  reduce 
all  losses  by  friction  and  shock.  The  section  of  the  crowns  of  the 
wheel  of  the  Boyden  turbine  is  shown  in  Fig.  199.  Outside  of 
the  turbine  wheel  was  fitted  a  "diffuser"  through  which,  after 
leaving  the  wheel,  the  water  moved  radially  with  a  continuously 
diminishing  velocity,  and  finally  entered  the  tail  race  with  a 
velocity  much  less,  than  if  it  had  done  so  direct  from  the  wheel. 
The  loss  by  velocity  head  was  thus  diminished,  and  Boyden 
claimed  that  the  diffuser  increased  the  efficiency  by  3  per  cent. 

186.     Some  actual  outward  flow  turbines. 

Double  outward  flow  turbines.  The  general  arrangement  of  an 
outward  flow  turbine  as  installed  at  Chevres  is  shown  in  Fig.  200. 
There  are  four  wheels  fixed  to  a  vertical  shaft,  two  of  which 
receive  the  water  from  below,  and  two  from  above.  The  fall 
varies  from  27  feet  in  dry  weather  to  14  feet  in  time  of  flood. 

The  upper  wheels  only  work  in  time  of  flood,  while  at  other 
times  the  full  power  is  developed  by  the  lower  wheels  alone,  the 
cylindrical  sluices  which  surround  the  upper  wheels  being  set  in 
such  a  position  as  to  cover  completely  the  exit  to  the  wheel. 

The  water  after  leaving  the  wheels,  diminishes  gradually  in 
velocity,  in  the  concrete  passages  leading  to  the  tail  race,  and  the 
loss  of  head  due  to  the  velocity  with  which  the  water  enters  the 

*  Lowell  Hydraulic  Experiments,  J.  B.  Francis,  1855. 


316 


HYDRAULICS 


tail  race  is  consequently  small.  These  passages  serve  the  same 
purpose  as  Boyden's  diffuser,  and  as  the  enlarging  suction  tube, 
in  that  they  allow  the  velocity  of  exit  to  diminish  gradually. 


Fig.  200.     Double  Outward  Flow  Turbine.     (Escher  Wyss  and  Co.) 

Outward  flow  turbine  with  horizontal  axis.  Fig.  201  shows  a 
section  through  the  wheel,  and  the  supply  and  exhaust  pipes,  of  an 
outward  flow  turbine,  having  a  horizontal  axis  and  exhausting 
down  a  "  suction  pipe."  The  water  after  leaving  the  wheel  enters 
a  large  chamber,  and  then  passes  down  the  exhaust  pipe,  the 
lower  end  of  which  is  below  the  tail  race. 

The  supply  of  water  to  the  wheel  is  regulated  by  a  horizontal 
cylindrical  gate  S,  between  the  guide  blades  Gr  and  the  wheel.  The 
gate  is  connected  to  the  ring  R,  which  slides  on  guides,  outside 
the  supply  pipe  P,  and  is  under  the  control  of  the  governor. 

The  pressure  of  the  water  in  the  supply  pipe  is  prevented  from 
causing  end  thrust  on  the  shaft  by  the  partition  T,  and  between 
T  and  the  wheel  the  exhaust  water  has  free  access. 

Outward  flow  turbines  at  Niagara  Falls.  The  first  turbines 
installed  at  Niagara  Falls  for  the  generation  of  electric  power, 


TURBINES 


317 


were  outward  flow  turbines  of  the  type  shown  in  Figs.  202  and 
203. 

There  are  two  wheels  on  the  same  vertical  shaft,  the  water 
being  brought  to  the  chamber  between  the  wheels  by  a  vertical 
penstock  7  6"  diameter.  The  water  passes  upwards  to  one  wheel 
and  downwards  to  the  other. 


Fig.  201.     Outward  Flow  Turbine  with  Suction  Tube. 

As  shown  in  Fig.  202  the  water  pressure  in  the  chamber  is 
prevented  from  acting  on  the  lower  wheel  by  the  partition  MN, 
but  is  allowed  to  act  on  the  lower  side  of  the  upper  wheel,  the 
upper  partition  HK  having  holes  in  it  to  allow  the  water  free  access 
underneath  the  wheel.  The  weight  of  the  vertical  shaft,  and  of 
the  wheels,  is  thus  balanced,  by  the  water  pressure  itself. 

The  lower  wheel  is  fixed  to  a  solid  shaft,  which  passes  through 
the  centre  of  the  upper  wheel,  and  is  connected  to  the  hollow 
shaft  of  the  upper  wheel  as  shown  diagrammatically  in  Fig.  202. 
Above  this  connection,  the  vertical  shaft  is  formed  of  a  hollow 


318 


HYDRAULICS 


tube  38  inches  diameter,  except  where  it  passes  through  the 
bearings,  where  it  is  solid,  and  11  inches  diameter. 

A  thrust  block  is  also  provided  to  carry  the  unbalanced 
weight. 

The  regulating  sluice  is  external  to  the  wheel.  To  maintain  a 
high  efficiency  at  part  gate,  the  wheel  is  divided  into  three  separate 
compartments  as  in  Fourneyron's  wheel. 


HoLUovr  and 
Solid  Shaft 

Water  cwLrrvilted, 
urui&r  the/  upper 
•wh&e/L  to  support 
the,  weight 
of  the,  shaft 


Fig.  202.     Diagrammatic  section  of  Outward  Flow  Turbine  at  Niagara  Falls. 

A  vertical  section  through  the  lower  wheel  is  shown  in  Fig. 
203,  and  a  part  sectional  plan  of  the  wheel  and  guide  blades  in 
Fig.  195. 

(Further  particulars  of  these  turbines  and  a  description  of  the 
governor  will  be  found  in  Cassier's  Magazine,  Yol.  III.,  and  in 
Turbines  Actuelle)  Buchetti,  Paris  1901. 

187.    Inward  flow  turbines. 

In  an  inward  flow  turbine  the  water  is  directed  to  the  wheel 
through  guide  passages  external  to  the  wheel,  and  after  flowing 
radially  finally  leaves  the  wheel  in  a  direction  parallel  to  the  axis. 

Like  the  outward  flow  turbine  it  may  work  drowned  or  with  a 
suction  tube. 

The  water  only  acts  upon  the  blades  during  the  radial 
movement. 


TUKBINES 


319 


As  improved  by  Francis*,  in  1849,  the  wheel  was  of  the  form 
shown  in  Fig.  204  and  was  called  by  its  inventor  a  "central  vent 
wheel." 


The  wheel  is  carried  on  a  vertical  shaft,  resting  on  a  footstep, 
and  supported  by  a  collar  bearing  placed  above  the  staging  S. 


*  Lowell  Hydraulic  Experiments,  F.  B.  Francis,  1855. 


320 


HYDRAULICS 


Above  the  wheel  is  a  heavy  casting  C,  supported  by  bolts 
from  the  staging  S,  which  acts  as  a  guide  for  the  cylindrical 
sluice  F,  and  carries  the  bearing  B  for  the  shaft.  There  are 
40  vanes  in  the  wheel  shown,  and  40  fixed  guide  blades,  the  former 
being  made  of  iron  one  quarter  of  an  inch  thick  and  the  latter 
three-sixteenths  of  an  inch. 


Fig.  204.     Francis'  Inward  flow  or  Central  vent  Turbine. 

The  triangles  of  velocities  at  inlet  and  outlet,  Fig.  205,  are 
drawn,  exactly  as  for  the  outward  flow  turbine,  the  only  difference 
being  that  the  velocities  vt  U,  Y,  Vr  and  u  refer  to  the  outer 


TURBINES 


321 


periphery,  and  Vi,  Ui,  Vi,  vr  and  u^  to  the  inner  periphery  of  the 
wheel. 

The  work  done  on  the  wheel  is 

—  ft.  Ibs.  per  lb., 
9         9 

and  neglecting  friction, 

VlJ  "Vl^l          TT          Ul2 

~g~  ~g~         ~~20' 

For  maximum  efficiency,  for  a  given  flow  through  the  wheel, 
Ui  should  be  radial  exactly  as  for  the  outward  flow  turbine. 


Fig.  205. 

The  student  should  work  the  following  example. 

The  outer  diameter  of  an  inward  flow  turbine  wheel  is  7'70  feet,  and  the  inner 
diameter  6-3  feet,  the  wheel  makes  55  revolutions  per  minute.  The  head  is 
14-8  feet,  the  velocity  at  inlet  is  25  feet  per  sec.,  and  the  radial  velocity  may  be 
assumed  constant  and  equal  to  7'5  feet.  Neglecting  friction,  draw  the  triangles  of 
velocities  at  inlet  and  outlet,  and  find  the  directions  of  the  tips  of  the  vanes  at 
inlet  and  outlet  so  that  there  may  be  no  shock  and  the  water  may  leave  radially. 

Loss  of  head  by  friction.  The  losses  of  head  by  friction  are 
similar  to  those  for  an  outward  flow  turbine  (see  page  313)  and 
the  general  formula  becomes 


9         9 
When  the  flow  is  radial  at  exit, 

—  =  eH. 
9 

The  value  of  e  varying  as  before  between  0'78  and  0'90. 

Example  (I).  An  inward  flow  turbine  working  under  a  head  of  80  feet  has 
radial  blades  at  inlet,  and  discharges  radially.  The  angle  the  tip  of  the  guide 
blade  makes  with  the  tangent  at  the  inlet  is  30  degrees  and  the  radial  velocity  is 
constant.  The  ratio  of  the  radii  at  inlet  and  outlet  is  1-75.  Find  the  velocity  of 
the  inlet  circumference  of  the  wheel.  Neglect  friction. 


L.  H. 


21 


322 


HYDRAULICS 


Since  the  discharge  is  radial,  the  velocity  at  exit  is 
'Ul  =  vl  tan  30° 

v 


1-75 


tan  30°. 


Then 


Vt?_  vz     tan2  30° 

~     '~ 


1-752       2g 
and  since  the  blades  are  radial  at  inlet  V  is  equal  to  v, 

v*     tan2  30° 
v2  =  5--8°-TT^2— o 


^      e 
therefore 

from4which 


/3 
V  =  Yr 


32  x  80 


=  49-3  ft.  per  sec. 


Fig.  206. 

Example  (2).  The  outer  diameter  of  the  wheel  of  an  inward  flow  turbine  of 
200  horse-power  is  2-46  feet,  the  inner  diameter  is  1-968  feet.  The  effective  width 
of  the  wheel  at  inlet  — 1 -15  feet.  The  head  is  39*5  feet  and  59  cubic  feet  of 
water  per  second  are  supplied.  The  radial  velocity  with  which  the  water  leaves 
the  wheel  may  be  taken  as  10  feet  per  second. 

Determine  the  theoretical  hydraulic  efficiency  E  and  the  actual  efficiency  el  of 
the  turbine,  and  design  suitable  vanes. 


200  x  550 
x  59x62-5 


"   5  /0> 


Theoretical  hydraulic  efficiency 


The  radial  velocity  of  flow  at  inlet, 

W=S-TT: 7-^  =  6-7  feet  per  sec. 

2-46  XTTX  1-15 


TUKBINES  323 

The  peripheral  velocity 

i?  =  2-46  .  TT  x  3^j°-  =  38-6  feet. 

The  velocity  of  whirl  V.     Assuming  a  hydraulic  efficiency  of  85°/0,  from 
the  formula 


9 
39-5  x  32-2  x  -85 


38-6 

=  28-0  feet  per  sec. 
The  angle  6.     Since  w  =  6'7  ft.  per  sec.  and  V  =  28'0  ft.  per  sec. 


tan  0  =       = 

0  =  13°  27'. 
The  angle  0.     Since  V  is  less  than  v,  0  is  greater  than  90°. 

tan  ^=-^=-^1=  -0-531, 

and  0  =  152°. 

For  the  water  to  discharge  radially  with  a  velocity  of  10  feet  per  sec. 


and  a  =18°  nearly. 

The  theoretical  vanes  are  shown  in  Fig.  206. 
Example  (3).     Find  the  values  of  0  and  a  on  the  assumption  that  e  is  0-80. 

Thomson's  inward  flow  turbine.  In  1851  Professor  James 
Thomson  invented  an  inward  flow  turbine,  the  wheel  of  which 
was  surrounded  by  a  large  chamber  set  eccentrically  to  the  wheel, 
as  shown  in  Figs.  207  to  210. 

Between  the  wheel  and  the  chamber  is  a  parallel  passage,  in 
which  are  four  guide  blades  Gr,  pivoted  on  fixed  centres  C  and 
which  can  be  moved  about  the  centres  C  by  bell  crank  levers, 
external  to  the  casing,  and  connected  together  by  levers  as  shown 
in  Fig.  207.  The  water  is  distributed  to  the  wheel  by  these  guide 
blades,  and  by  turning  the  worm  quadrant  Q  by  means  of  the 
worm,  the  supply  of  water  to  the  wheel,  and  thus  the  power  of 
the  turbine,  can  be  varied.  The  advantage  of  this  method  of 
regulating  the  flow,  is  that  there  is  no  sudden  enlargement  from 
the  guide  passages  to  the  wheel,  and  the  efficiency  at  part  load 
is  not  much  less  than  at  full  load. 

Figs.  209  and  210  show  an  enlarged  section  and  part  sectional 
elevation  of  the  turbine  wheel,  and  one  of  the  guide  blades  Gr. 
The  details  of  the  wheel  and  casing  are  made  slightly  different 
from  those  shown  in  Figs.  207  and  208  to  illustrate  alternative 
methods. 

The  sides  or  crowns  of  the  wheel  are  tapered,  so  that  the 
peripheral  area  of  the  wheel  at  the  discharge  is  equal  to  the 
peripheral  area  at  inlet.  The  radial  velocities  of  flow  at  inlet 
and  outlet  are,  therefore,  equal. 

21—2 


324 


HYDRAULICS 


The  inner  radius  r  in  Thomson's  turbine,  and  generally  in 
turbines  of  this  class  made  by  English  makers,  is  equal  to  one-half 
the  external  radius  R. 


Fig.  207.     Guide  blades  and  casing  of  Thomson  Inward  Flow  Turbine. 

The  exhaust  for  the  turbine  shown  takes  place  down  two 
suction  tubes,  but  the  turbine  can  easily  be  adapted  to  work  below 
the  tail  water  level. 

As  will  be  seen  from  the  drawing  the  vanes  of  the  wheel  are 
made  alternately  long  and  short,  every  other  one  only  continuing 
from  the  outer  to  the  inner  periphery. 


TURBINES 


325 


The  triangles  of  velocities  for  the  inlet  and  outlet  are  shown  in 
Fig.  211,  the  water  leaving  the  wheel  radially. 

The  path  of  the  water  through  the  wheel,  relative  to  the  fixed 
casing,  is  also  shown  and  was  obtained  by  the  method  described 
on  page  312. 

Inward  flow  turbines  with  adjustable  guide  blades,  as  made  by 
the  continental  makers,  have  a  much  greater  number  of  guide 
blades  (see  Fig.  233,  page  352). 


Fig.  208.      Section  through  wheel  and  casing  of  Thomson  Inward  Flow  Turbine. 

188.     Some  actual  inward  flow  turbines. 

A  later  form  of  the  Francis  inward  flow  turbine  as  designed  by 
Pictet  and  Co.,  and  having  a  horizontal  shaft,  is  shown  in  Fig.  212. 

The  wheel  is  double  and  is  surrounded  by  a  large  chamber 
from  which  water  flows  through  the  guides  Gr  to  the  wheel  W. 
After  leaving  the  wheel,  exhaust  takes  place  down  the  two  suction 
tubes  S,  thus  allowing  the  turbine  to  be  placed  well  above  the 
tail  water  while  utilising  the  full  head. 

The  regulating  sluice  F  consists  of  a  steel  cylinder,  which 
slides  in  a  direction  parallel  to  the  axis  between  the  wheel  and 
guides. 


HYDRAULICS 


Gwicie 


Fig.  209.  Fig.  210. 

Detail  of  wheel  and  guide  blade  of  Thomson  Inward  Flow  Turbine. 


Fig.  211. 


TURBINES 


327 


The  wheel  is  divided  into  five  separate  compartments,  so  that 
at  any  time  only  one  can  be  partially  closed,  and  loss  of  head  by 
contraction  and  sudden  enlargement  of  the  stream,  only  takes 
place  in  this  one  compartment. 


328 


HYDRAULICS 


The  sluice  F  is  moved  by  two  screws  T,  which  slide  through 
stuffing  boxes  B,  and  which  can  be  controlled  by  hand  or  by  the 
governor  B,. 

Inward  flow  turbine  for  low  falls  and  variable  head.  The 
turbine  shown  in  Fig.  213  is  an  example  of  an  inward  flow  turbine 
suitable  to  low  falls  and  variable  head.  It  has  a  vertical  axis  and 
works  drowned.  The  wheel  and  the  distributor  surrounding  the 
wheel  are  divided  into  five  stages,  the  two  upper  stages  being 
shallower  than  the  three  lower  ones,  and  all  of  which  stages  can 


,,.C     1Z34-S67S      9FE£T 


Fig.  213.    Inward  Flow  Turbine  for  a  low  and  variable  fall.     (Pictet  and  Co. 


TURBINES  329 

be  opened  or  closed  as  required  by  the  steel  cylindrical  sluice  CC 
surrounding  the  distributor. 

When  one  of  the  stages  is  only  partially  closed  by  the  sluice, 
a  loss  of  efficiency  must  take  place,  but  the  efficiency  of  this  one 
stage  only  is  diminished,  the  stages  that  are  still  open  working 
with  their  full  efficiency.  With  this  construction  a  high  efficiency 
of  the  turbine  is  maintained  for  partial  flow.  With  normal  flows, 
and  a  head  of  about  6'25  feet,  the  three  lower  stages  only  are 
necessary  to  give  full  power,  and  the  efficiency  is  then  a 
maximum.  In  times  of  flood  there  is  a  large  volume  of  water 
available,  but  the  tail  water  rises  so  that  the  head  is  only  about 
4'9  feet,  the  two  upper  stages  can  then  be  brought  into  operation 
to  accommodate  a  larger  flow,  and  thus  the  same  power  may  be 
obtained  under  a  less  head.  The  efficiency  is  less  than  when  the 
three  stages  only  are  working,  but  as  there  is  plenty  of  water 
available,  the  loss  of  efficiency  is  not  serious. 

The  cylinder  C  is  carried  by  four  vertical  spindles  S,  having 
racks  R  fixed  to  their  upper  ends.  G-earing  with  these  racks,  are 
pinions  p,  Fig.  213,  all  of  which  are  worked  simultaneously  by  the 
regulator,  or  by  hand.  A  bevel  wheel  fixed  to  the  vertical  shaft 
gears  with  a  second  bevel  wheel  on  a  horizontal  shaft,  the  velocity 
ratio  being  3  to  1. 

189.  The  best  peripheral  velocity  for  inward  and  outward 
flow  turbines. 

When  the  discharge  is  radial,  the  general  formula,  as  shown  on 
page  315,  is 

—  =  eH  =  0-78toO'90H..  ...(1). 

9 

If  the  blades  are  radial  at  inlet,  for  no  shock,  v  should  be  equal 
to  Y,  and 


or  v  =  V  -  0'624  to  0'67  \20H. 

This  is  sometimes  called  the  best  velocity  for  v,  but  it  should  be 
clearly  understood  that  it  is  only  so  when  the  blades  are  radial  at 
inlet. 

190.  Experimental  determination  of  the  best  peripheral 
velocity  for  inward  and  outward  flow  turbines. 

For  an  outward  flow  turbine,  working  under  a  head  of  14  feet, 
with  blades  radial  at  inlet,  Francis  *  found  that  when  v  was 

0-626 


*  Lowell,  Hydraulic  Experiments. 


330  HYDRAULICS 

the  efficiency  was  a  maximum  and  equal  to  79'37  per  cent.  The 
efficiency  however  was  over  78  per  cent,  for  all  values  of  v 
between  0*545  \/2^H  and  "671  \/20H.  If  3  per  cent,  be  allowed 
for  the  mechanical  losses  the  hydraulic  efficiency  may  be  taken 
as  82*4  per  cent. 

Yv 

From  the  formula  —  =  *824H,  and  taking  V  equal  to  v, 


v  =  '64 
so  that  the  result  of  the  experiment  agrees  well  with  the  formula. 

For  an  inward  flow  turbine  having  vanes  as  shown  in  Fig.  205, 
the  total  efficiency  was  over  79  per  cent,  for  values  of  v  between 
0*624  \/2^H  and  0*708  *J2gK,  the  greatest  efficiency  being  797 
per  cent,  when  v  was  0'708  \/20H  and  again  when  v  was 
•637  s/2^H. 

It  will  be  seen  from  Fig.  205  that  although  the  tip  of  the  vane 
at  the  convex  side  is  nearly  radial,  the  general  direction  of  the 
vane  at  inlet  is  inclined  at  an  angle  greater  than  90  degrees  to 
the  direction  of  motion,  and  therefore  for  no  shock  V  should  be 
less  than  v. 

When  v  was  '708  \/2#H,  V,  Fig.  205,  was  less  than  v.  The 
value  of  Y  was  deduced  from  the  following  data,  which  is  also 
useful  as  being  taken  from  a  turbine  of  very  high  efficiency. 

Diameter  of  wheel  9*338  feet. 

Width  between  the  crowns  at  inlet  0*999  foot. 

There  were  40  vanes  in  the  wheel  and  an  equal  number  of 
fixed  guides  external  to  the  wheel. 

The  minimum  width  of  each  guide  passage  was  0*1467  foot  and 
.the  depth  1*0066  feet. 

The  quantity  of  water  supplied  to  the  wheel  per  second  was 
112*525  cubic  feet,  and  the  total  fall  of  the  water  was  13*4  feet. 
The  radial  velocity  of  flow  u  was,  therefore,  3*86  feet  per  second. 

The  velocity  through  the  minimum  section  of  the  guide  passage 
was  19  feet  per  second. 

When  the  efficiency  was  a  maximum,  v  was  20*8  feet  per  sec. 
Then  the  radial  velocity  of  flow  at  inlet  to  the  wheel  being 
3*86  feet,  and  U  being  taken  as  19  feet  per  second,  the  triangle 
of  velocities  at  inlet  is  ABC,  Fig.  205,  and  Y  is  18*4  feet  per  sec. 

If  it  is  assumed  that  the  water  leaves  the  wheel  radially,  then 

eH  =  —  =  11*85  feet. 
9 

1  1  'Q£\ 

The  efficiency  e  should  be  ..~,  =  88*5  per  cent.,  which  is  9  per 
cent,  higher  than  the  actual  efficiency. 


TURBINES  331 

The  actual  efficiency  however  includes  not  only  the  fluid  losses 
but  also  the  mechanical  losses,  and  these  would  probably  be  from 
2  to  8  per  cent.,  and  the  actual  work  done  by  the  turbine  on  the 
shaft  is  probably  between  80  and  86*5  per  cent,  of  the  work  done 
by  the  water. 

Vv 
191.     Value  of  e  to  be  used  in  the  formula  —  =  eH. 

g 

~Vv 

In  general,  it  may  be  said  that,  in  using  the  formula  —  =  eH, 

y 

the  value  of  e  to  be  used  in  any  given  case  is  doubtful,  as  even 
though  the  efficiency  of  the  class  of  turbines  may  be  known,  it  is 
difficult  to  say  exactly  how  much  of  the  energy  is  lost  mechanically 
and  how  much  hydraulically. 

A  trial  of  a  turbine  without  load,  would  be  useless  to  deter- 
mine the  mechanical  efficiency,  as  the  hydraulic  losses  in  such  a 
trial  would  be  very  much  larger  than  when  the  turbine  is  working 
at  full  load.  By  revolving  the  turbine  without  load  by  means  of 
an  electric  motor,  or  through  the  medium  of  a  dynamometer,  the 
work  to  overcome  friction  of  bearings  and  other  mechanical  losses 
could  be  found.  At  all  loads,  from  no  load  to  full  load,  the 
frictional  resistances  of  machines  are  fairly  constant,  and  the 
mechanical  losses  for  a  given  class  of  turbines,  at  the  normal  load 
for  which  the  vane  angles  are  calculated,  could  thus  approximately 
be  obtained.  If,  however,  in  making  calculations  the  difference 
between  the  actual  and  the  hydraulic  efficiency  be  taken  as,  say, 
5  per  cent.,  the  error  cannot  be  very  great,  as  a  variation  of  5  per 
cent,  in  the  value  assumed  for  the  hydraulic  efficiency  e,  will  only 
make  a  difference  of  a  few  degrees  in  the  calculated  value  of 
the  angle  <£. 

The  best  value  for  e,  for  inward  flow  turbines,  is  probably  0'80, 
and  experience  shows  that  this  value  may  be  used  with  confidence. 

Example.     Taking  the  data  as  given  in  the  example  of  section  184,  and  assuming 
an  efficiency  for  the  turbine  of  75  per  cent.,  the  horse-power  is 
._     215  x  62-4  x  141-5  x  -75  x  60 

33,000 

=  2600  horse-power. 

If  the  hydraulic   efficiency  is   supposed   to  be  80   per  cent.,   the  velocity  of 
whirl  V  should  be 

.H     0-8.32.141-5 


•y 


Then 


v  69 

=  52  feet  per  sec. 
18-35        -18-35 


and  0  =  132°  47'. 

Now  suppose  the  turbine  to  be  still  generating  2600  horse-power,  and  to  have 
an  efficiency  of  80  per  cent.,  and  a  hydraulic  efficiency  of  85  per  cent. 


332 


HYDRAULICS 


Then  the  quantity  of  water  required  per  second,  is 
215x0-75 


Q=- 


0-8 


=  200  cubic  feet  per  sec. 


and  the  radial  velocity  of  flow  at  inlet  will  be 
18-35x200 


215 
•85.32.  141-5 


=  17*1  ft.  per  sec. 


Then 


tan  <f>  = 


69 
17-1 


:55'4  ft.  per  sec. 


-17-1 


55-4-69"    13-6 


=  128°.  24'. 


192.     The  ratio  of  the  velocity  of  whirl  V  to  the  velocity 
of  the  inlet  periphery  v. 

Experience  shows  that,  consistent  with.  Vv  satisfying  the  general 

formula,  the  ratio  ^  may  vary  between  very  wide  limits  without 
considerably  altering  the  efficiency  of  the  turbine. 


Table  XXXYII  shows  actual  values  of  the  ratio 


taken 


from  a  number  of  existing  turbines,  and  also  corresponding  values 


Fig.  214. 


TURBINES 


333 


yv 
,  V  being  calculated  from  —  =  0*8H.    The  corresponding 

variation  in  the  angle  <£,  Fig.  214,  is  from  20  to  150  degrees. 

For  a  given  head,  v  may  therefore  vary  within  wide  limits, 
which  allows  a  very  large  variation  in  the  angular  velocity  of  the 
wheel  to  suit  particular  circumstances. 

TABLE  XXXVII. 

Showing  the  heads,  and  the  velocity  of  the  receiving  circum- 
ference v  for  some  existing  inward  and  outward,  and  mixed  flow 
turbines. 


Ratio 

v 

Ratio 

Hfeet 

v  feet 
per  sec. 

v/2.9H 

v 

H.P. 

V  being  calculated 
Vv 

/9^H 

v  &gti 

from  —  =  -8H 

9 

Inward  flow  : 

Niagara  Falls* 

146 

70 

96-8 

0-72 

5000 

0-555 

Rheinfelden 

14-8 

22 

30-7 

0-71 

840 

0-565 

By  Theodor  ) 
Bell  and  Co.  J" 

28-4 

39 

42-6 

0-91 

0-44 

60-4 

32-2 

62-3 

0-52 

0-77 

Pictet  and  Co. 

183-7 

51-1 

76-8 

0-47 

300 

0-85 

M 

134-5 

46-6 

65-6 

0-505 

300 

0-79 

6-25 

16-6 

20 

0-83 

0-48 

11 

30 

25-75 

44 

0-58 

700 

0-69 

11 

38-5 

50-3 

0-77 

200 

0-52 

Ganz  and  Co. 

112 

64-3 

84-6 

0-54 

0-74 

225 

64-7 

120 

0-54 

682 

0-58 

Rioter  and  Co. 

10-66 

15-2 

26 

0-585 

30 

0-69 

Outward  flow  : 

Niagara  Falls 

141-5 

69 

95-2 

0-725) 

rnnn 

0-55 

Pictet  and  Co. 

130-5 

69 

91-6 

0-750) 

O\J\J\J 

0-53 

Ganz  and  Co. 

95-1 

38-7 

78-0 

0-495 

290 

0-81 

» 

223 

55-6 

120-0 

0-46 

1200 

0-87 

*  Escher  Wyss  and  Co. 

For  example,  if  a  turbine  is  required  to  drive  alternators 
direct,  the  number  of  revolutions  will  probably  be  fixed  by  the 
alternators,  while,  as  shown  later,  the  diameter  of  the  wheel  is 
practically  fixed  by  the  quantity  of  water,  which  it  is  required  to 
pass  through  the  wheel,  consistent  with  the  peripheral  velocity  of 
the  wheel,  not  being  greater  than  100  feet  per  second,  unless,  as 
in  the  turbine  described  on  page  373,  special  precautions  are 
taken.  This  latter  condition  may  necessitate  the  placing  of  two 
or  more  wheels  on  one  shaft. 


334  HYDRAULICS 

Suppose  then,  the  number  of  revolutions  of  the  wheel  to  be 
given  and  d  is  fixed,  then  v  has  a  definite  value,  and  V  must  be 
made  to  satisfy  the  equation 

V*      R 
—  =  ei±. 

9 

Fig.  214  is  drawn  to  illustrate  three  cases  for  which  Yy  is 
constant.  The  angles  of  the  vanes  at  outlet  are  the  same  for  all 
three,  but  the  guide  angle  0  and  the  vane  angle  <£  at  inlet  vary 
considerably. 

193.  The  velocity  with  which  water  leaves  a  turbine. 
In  a  well-designed  turbine  the  velocity  with  which  the  water 

leaves  the  turbine  should  be  as  small  as  possible,  consistent  with 
keeping  the  turbine  wheel  and  the  down-take  within  reasonable 
dimensions. 

In  actual  turbines  the  head  lost  due  to  this  velocity  head 
varies  from  2  to  8  per  cent.  If  a  turbine  is  fitted  with  a 
suction  pipe  the  water  may  be  allowed  to  leave  the  wheel  itself 
with  a  fairly  high  velocity  and  the  discharge  pipe  can  be  made 
conical  so  as  to  allow  the  actual  discharge  velocity  to  be  as  small 
as  desired.  It  should  however  be  noted  that  if  the  water  leaves 
the  wheel  with  a  high  velocity  it  is  more  than  probable  that  there 
will  be  some  loss  of  head  due  to  shock,  as  it  is  difficult  to  ensure 
that  water  so  discharged  shall  have  its  velocity  changed  gradually. 

194.  Bernouilli's  equations  applied  to  inward  and  out- 
ward flow  turbines  neglecting  friction. 

Centrifugal  head  impressed  on  the  water  by  the  wheel.  The 
theory  of  the  reaction  turbines  is  best  considered  from  the  point 
of  view  of  Bernoulli's  equations ;  but  before  proceeding  to  discuss 
them  in  detail,  it  is  necessary  to  consider  the  "  centrifugal  head  " 
impressed  on  the  water  by  the  wheel. 

This  head  has  already  been  considered  in  connection  with  the 
Scotch  turbine,  page  303. 

Let  r,  Fig.  216,  be  the  internal  radius  of  a  wheel,  and  E  the 
external  radius. 

At  the  internal  circumference  let  the  wheel  be  covered  with  a 
cylinder  c  so  that  there  can  be  no  flow  through  the  wheel,  and  let 
it  be  supposed  that  the  wheel  is  made  to  revolve  at  the  angular 
velocity  u>  which  it  has  as  a  turbine,  the  wheel  being  full  of  water 
and  surrounded  by  water  at  rest,  the  pressure  outside  the  wheel 
being  sufficient  to  prevent  the  water  being  whirled  out  of  the 
wheel.  Let  d  be  the  depth  of  the  wheel  between  the  crowns. 
Consider  any  element  of  a  ring  of  radius  r0  and  thickness  dr,  and 
subtending  a  small  angle  0  at  the  centre  C,  Fig.  216. 


TURBINES 
The  weight  of  the  element  is 


335 


.dr.d, 
and  the  centrifugal  force  acting  on  the  element  is 


00  .dr.d.  w 


-°  Ibs. 


Let  p  be  the  pressure  per  unit  area  on  the  inner  force  of  the 
element  and  p  +  dp  on  the  outer. 

wr06  .dr.d.  wV0 


Then 


.  d 


PC 


m 


PC 


Fig.  215. 


Fig.  216. 


The  increase  in  the  pressure,  due  to  centrifugal  forces,  between 
r  and  R  is,  therefore, 


(Rw<o2  w 

—  tvZr.= 


and 


For  equilibrium,  therefore,  the  pressure  in  the  water  surround- 
ing the  wheel  must  be  pc. 

If  now  the  cylinder  c  be  removed  and  water  is  allowed  to  flow 
through  the  wheel,  either  inwards  or  outwards,  this  centrifugal 
head  will  always  be  impressed  upon  the  water,  whether  the  wheel 
is  driven  by  the  water  as  a  turbine,  or  by  some  external  agency, 
and  acts  as  a  pump. 

Bernoulli? s  equations.  The  student  on  first  reading  these 
equations  will  do  well  to  confine  his  attention  to  the  inward  flow 
turbine,  Fig.  217,  and  then  read  them  through  again,  confining  his 
.attention  to  the  outward  flow  turbine,  Fig.  191. 


.336 


HYDRAULICS 


Let  p  be  the  pressure  at  A,  the  inlet  to  the  wheel,  or  in  the 
clearance  between  the  wheel  and  the  guides,  pi  the  pressure  at 
the  outlet  B,  Fig.  217,  and  pa  the  atmospheric  pressure,  in  pounds 
per  square  foot.  Let  H  be  the  total  head,  and  H0  the  statical 
head  at  the  centre  of  the  wheel.  The  triangles  of  velocities  are 
as  shown  in  Figs.  218  and  219. 

Then  at  A 


w 


w 


CD. 


Between  B  and  A  the  wheel  impresses  upon  the  water  the 
centrifugal  head 


v  being  greater  than 
outward  flow. 


_ 

2<7     20' 
for  an  inward  flow  turbine  and  less  for  the 


Fig.  217. 


Consider  now  the  total  head  relative  to  the  wheel  at  A  and  B. 

V  2  p 

The  velocity  head  at  A  is  --  and  the  pressure  head  is  —  ,  and 


at  B  the  velocity  and  pressure  heads  are  ^j-  and  —  respectively. 

If  no  head  were  impressed  on  the  water  as  it  flows  through 
the  wheel,  the  pressure  head  plus  the  velocity  head  at  A  and  B 
would  be  equal  to  each  other.  But  between  A  and  B  there  is 
impressed  on  the  water  the  centrifugal  head,  and  therefore, 


w      20  '  20     20     w      20 


TURBINES  337 

This  equation  can  be  used  to  deduce  the  fundamental  equation, 

*    ™  V  1^1   _   7  ,^, 

y~  ~g~~h (3)* 

From  the  triangles  ODE  and  ADE,  Fig.  218, 

Yr2  -  (Y  -  v)2  +  u2  and  Y2  +  w2  =  U2, 
and  from  the  triangle  BFGr,  Fig  219, 

vr2  =  (vi  -  Yi)2  +  u2  and  Yi2  +  u?  =  Ui2. 
Therefore  by  substitution  in  (2), 

Pi      fa-YQ2     v*__vl     UL  =  P_     (V-vY     u? 

w  2g  2g     2g+  2g     w+       2g      +  2g    -(4!)' 


From  which 


UI'  =  P  ,  u2   ^v 

2g      w     2g       g  ' 


and 


v~V     vl"V1=p     P!  |  U2     W 
g         g       w      w     2g      2g 

Substituting  for  P  +  5?  from  (1) 


_  -  TT    u.      «  i  l 

—  -do  "•" 

g        g  w     w     2g 


(5). 


.(6). 


Fig.  218. 


Fig.  219. 


Wheel  in  suction  tube.  If  the  centre  of  the  wheel  is  7i0  feet 
above  the  surface  of  the  tail  water,  and  U0  is  the  velocity  with 
which  the  water  leaves  the  down-pipe,  then 


w 


Substituting  for  —  +  ~  in  (6), 


^-^=H0+^-&^0-ni2 

g         g  w     w  2g 

=  H-?^. 


L.    H. 


22 


338  HYDRAULICS 

IfVisO,  ?  =  H-1K 

The  wheel  can  therefore  take  full  advantage  of  the  head  H 
even  though  it  is  placed  at  some  distance  above  the  level  of  the 
tail  water. 

Drowned  wheel.  If  the  level  of  the  tail  water  is  CD,  Fig.  217, 
or  the  wheel  is  drowned,  and  hi  is  the  depth  of  the  centre  of  the 
wheel  below  the  tail  race  level, 

fe-fc+fc, 

w  w 

and  the  work  done  on  the  wheel  per  pound  of  water  is  again 
vV 


g       g  g 

«Y 

IfViisO,  —  =  &. 

From  equation  (5), 

VV  _  Pi  Y!  =  p  _  Pi  +  IP  _  Ui2 

g         g        w     w      2g      2g  ' 

so  that  the  work  done  on  the  wheel  per  pound  is  the  difference 
between  the  pressure  head  plus  the  velocity  head  at  entrance  and 
the  pressure  head  plus  velocity  head  at  exit. 

In  an  impulse  turbine  p  and  pl  are  equal,  and  the  work  done 
is  then  the  change  in  the  kinetic  energy  of  the  jet  when  it  strikes 
and  when  it  leaves  the  wheel. 

A  special  €ase  arises  when  pi  is  equal  to  p.  In  this  case  a 
considerable  clearance  may  be  allowed  between  the  wheel  and  the 
fixed  guide  without  danger  of  leakage. 

Equation  (2),  for  this  case,  becomes 


and  if  at   exit  vr  is   made   equal   to  v1}  or  the   triangle  BFGr, 
Fig.  219,  is  isosceles, 


and  the  triangle  of  velocities  at  entrance  is  also  isosceles. 
The  pressure  head  at  entrance  is 


and  at  exit  is  either         —  +  hi ,  or  —  —  h0. 
w  w 


TURBINES  339 

Therefore,  since  the  pressures  at  entrance  and  exit  are  equal, 

U2     TT       I      TT 
2~  -  Mo  -  fii  =  J±, 

or  else  H0  +  ho  =  H. 

The  water  then  enters  the  wheel  with  a  velocity  equal  to  that 
due  to  the  total  head  H,  and  the  turbine  becomes  a  free-deviation 
or  impulse  turbine. 

195.  Bernouilli's  equations  for  the  inward  and  outward 
flow  turbines  including  friction. 

If  H/  is  the  loss  of  head  in  the  penstock  and  guide  passages, 
h/  the  loss  of  head  in  the  wheel,  he  the  loss  at  exit  from  the  wheel 
and  in  the  suction  pipe,  and  Ui  the  velocity  of  exhaust, 

P  +  ^!=  £_„_ 

w     2  w 


Pi  +  vl  +  ^L  _  ?L  =  P  +  Zn2_  T,  (9\ 

w      2g     2g     2g     w      2g       f  


and 


from  which  —  =  H-(^  +  fc,+  Hr+fc»)  ..................  (4). 

If  the  losses  can  be  expressed  as  a  fraction  of  H,  or  equal  to  KH, 
then 

V« 

—  =  (l-K)H  =  eH 

-078H  to  0'90H*. 

196.     Turbine  to  develop  a  given  horse-power. 

Let  H  be  the  total  head  in  feet  under  which  the  turbine  works. 

Let  n  be  the  number  of  revolutions  of  the  wheel  per  minute. 

Let  Q  be  the  number  of  cubic  feet  of  water  per  second  required 
by  the  turbine. 

Let  E  be  the  theoretical  hydraulic  efficiency. 

Let  e  be  the  hydraulic  efficiency. 

Let  em  be  the  mechanical  efficiency. 

Let  ei  be  the  actual  efficiency  including  mechanical  losses. 

Let  Ui  be  the  radial  velocity  with  which  the  water  leaves  the 
wheel. 

Let  D  be  the  diameter  of  the  wheel  in  feet  at  the  inlet  circum- 
ference and  d  the  diameter  at  the  outlet  circumference. 

Let  B  be  the  width  of  the  wheel  in  feet  between  the  crowns 
at  the  inlet  circumference,  and  b  be  the  width  between  the  crowns 
at  the  outlet  circumference. 

Let  N  be  the  horse-power  of  the  turbine. 
*  See  page  315. 

22—2 


340  HYDRAULICS 

The  number  of  cubic  feet  per  second  required  is 

N.  33,000 


eiH.62'4.60  ' 
A  reasonable  value  for  ei  is  75  per  cent. 
The  velocity  U0  with  which  the  water  leaves  the  turbine,  since 


is  U0=>/20(l-E)Hft.  per  sec (2). 

If  it  be  assumed  that  this  is  equal  to  Ui,  which  would  of 
necessity  be  the  case  when  the  turbine  works  drowned,  or 
exhausts  into  the  air,  then,  if  t  is  the  peripheral  thickness  of  the 
vanes  at  outlet  and  m  the  number  of  vanes, 


If  U0  is  not  equal  to  u1}  then 

(ird  -  mt)  u\  b  =  Q  (3) . 

The  number  of  vanes  m  and  the  thickness  t  are  somewhat 
arbitrary,  but  in  well-designed  turbines  t  is  made  as  small  as 
possible. 

As  a  first  approximation  mt  may  be  taken  as  zero  and  (3) 
becomes 

vdbu^Q  (4). 

For  an  inward  flow  turbine  the  diameter  d  is  fixed  from 
consideration  of  the  velocity  with  which  the  water  leaves  the 
wheel  in  an  axial  direction. 

If  the  water  leaves  at  both  sides  of  the  wheel  as  in  Fig.  208, 
and  the  diameter  of  the  shaft  is  d0,  the  axial  velocity  is 

UQ  = — ft.  per  sec. 

2  —  (^2  -  d 2) 

The  diameter  d0  can  generally  be  given  an  arbitrary  value,  or 
for  a  first  approximation  to  d  it  may  be  neglected,  and  UQ  may  be 
taken  as  equal  to  %.  Then 

d  = 


From  (4)  and  (5)  b  and  d  can  now  be  determined. 

A  ratio  for  -T  having  been  decided  upon,  D  can  be  calculated, 

and  if  the  radial  velocity  at  inlet  is  to  be  the  same  as  at  outlet, 
and  t0  is  the  thickness  of  the  vanes  at  inlet, 

(7rD-m£o)B=SL(7rd-mO&   (6). 


TURBINES  341 

For  rolled  brass  or  wrought  steel  blades,  t0  may  be  very  small, 
and  for  blades  cast  with  the  wheel,  by  shaping  them  as  in  Fig.  227, 
t0  is  practically  zero.  Then 

T*  _      Q 

~^D' 

If  now  the  number  of  revolutions  is  fixed  by  any  special 
condition,  such  as  having  to  drive  an  alternator  direct,  at  some 
definite  speed,  the  peripheral  velocity  is 

TrDfi  p,  m, 

V  =  -QQ-  ft-  per  sec  .........................  (7). 

V« 

Then  ^T  =  eH> 

y 

and  if  e  is  given  a  value,  say  80  per  cent., 

V  =  '^5  ft.  per  sec.  ...(8). 

v 

Since  u,  V,  and  v  are  known,  the  triangle  of  velocities  at  inlet 
can  be  drawn  and  the  direction  of  flow  and  of  the  tip  of  vanes 
at  inlet  determined.  Or  0  and  </>,  Fig.  214,  can  be  calculated  from 

tan^       ..............................  (9) 


and  tan<f>-~v    ........................  (10). 

Then  U,  the  velocity  of  flow  at  inlet,  is 


.,  „ 

At  exit  Vi  =  -QQ  ft.  per  sec., 

and  taking  u\  as  radial  and  equal  to  u,  the  triangle  of  velocities 
can  be  drawn,  or  a  calculated  from 

u 
tan  a  =  —  . 

Vi 

If  Ho  is  the  head  of  water  at  the  centre  of  the  wheel  and  H/  the 
head  lost  by  friction  in  the  supply  pipe  and  guide  passages,  the 
pressure  head  at  the  inlet  is 

P       TT        U'2      TT 
=  ±±o  —  o  --  **/• 

w  2g 

Example.     An  inward  flow  turbine  is  required  to  develop  300  horse-power  under 
a  head  60  feet,  and  to  run  at  250  revolutions  per  minute. 
To  determine  the  leading  dimensions  of  the  turbine. 
Assuming  el  to  be  75  per  cent., 

300  x  33,000 
^  ~  -75  x  60  x  62-4  x  60 
=  58-7  cubic  feet  per  sec. 


342  HYDRAULICS 

Assuming  E  is  95  per  cent.,  or  five  per  cent,  of  the  head  is  lost  by  velocity 
of  exit  and  u^=u, 

I  =  -05.  60 

and  u  =  13-8  feet  per  sec. 

Then  from  (5),  page  340, 


W2.1-36 


=  1-65  feet, 
say  20  inches  to  make  allowance  for  shaft  and  to  keep  even  dimension. 

Then  from  (4),  6  =  ^  =  -82  foot 

=  9£  inches  say. 
Taking  ?  as  1-8,  D  =  3-0  feet,  and 

v  =  TT  .  3  .  *#$•  =  39-3  feet  per  sec., 
and  B  =  5£  inches  say. 

Assuming  e  to  be  80  per  cent., 

8a=8(H)fl 

13-8 


and  0=19°  30', 

13-8 


and  0=91°  15'. 

13-8x1-8 


and  a  =  32°  18'. 

The  velocity  U  at  inlet  is 

U  =  N/39-02+(13-8)2 
=  41-3  ft.  per  sec. 
The  absolute  pressure  head  at  the  inlet  to  the  wheel  is 


-  ^  --  - 


-  --  f 


hf,  the  head  lost  by  friction  in  the  down  pipe 


The  pressure  head  at  the  outlet  of  the  wheel  will  depend  upon  the  height  of  the 
wheel  above  or  below  the  tail  water. 

197.    Parallel  or  axial  flow  turbines. 

Fig.  220  shows  a  double  compartment  axial  flow  turbine,  the 
guide  blades  bein  g  placed  above  the  wheel  and  the  flow  through 
the  wheel  being  parallel  to  the  axis.  The  circumferential  section 
of  the  vanes  at  any  radius  when  turned  into  the  plane  of  the 
paper  is  as  shown  in  Fig.  221.  A  plan  of  the  wheel  is  also  shown. 

The  triangles  of  velocities  at  inlet  and  outlet  for  any  radius 
are  similar  to  those  for  inward  and  outward  flow  turbines,  the 
velocities  v  and  vi,  Figs.  222  and  223,  being  equal. 


TUKBINES 

The  general  formula  now  becomes 


.343 


For  maximum  efficiency  for  a  given  flow,  the  water  should 
leave  the  wheel  in  a  direction  parallel  to  the  axis,  so  that  it  has 
no  momentum  in  the  direction  of  v. 


Fig.  220.     Double  Compartment  Parallel  Flow  Turbine. 


Figs.  221,   222,  22?. 

Then,  taking  friction  and  other  losses  into  account, 

Vv  „ 
-  =eH. 
9 


344 


HYDRAULICS 


The  velocity  v  will  be  proportional  to  the  radius,  so  that  if  the 
water  is  to  enter  and  leave  the  wheel  without  shock,  the  angles  0, 
<f>,  and  a  must  vary  with  the  radius. 

The  variation  in  the  form  of  the  vane  with  the  radius  is  shown 
by  an  example. 

A  Jonval  wheel  has  an  internal  diameter  of  5  feet  and  an 
external  diameter  of  8' 6".  The  depth  of  the  wheel  is  7  inches. 
The  head  is  15  feet  and  the  wheel  makes  55  revolutions  per 
minute.  The  flow  is  300  cubic  feet  per  second. 

To  find  the  horse-power  of  the  wheel,  and  to  design  the  wheel 
vanes. 

Let  TI  be  the  mean  radius,  and  r  and  r2  the  radii  of  the  wheel 
at  the  inner  and  outer  circumference  respectively.  Then 

r  =  2"5  feet      and  v  =  2irr  ££  =  14*4  feet  per  sec., 
TI  =  3"375  feet  and  Vi  =  2nri  J-£  =  21*5  feet  per  sec., 
r2  =  4'25  feet    and  v2  =  27rr2  f-J  =  24"  5  feet  per  sec. 
The  mean  axial  velocity  is 

300 

u=    7~2 — 1\  =0  15  ft.  per  sec. 
TT  (r,1  -  r2) 


y  I 


B  D 

Fig.  224.     Triangles  of  velocities  at  inlet  and  outlet  at  three  different 
•     radii  of  a  Parallel  Flow  Turbine. 


Taking  e  as  0'80  at  each  radius, 
0-8.32-2.15     385 


ft.  per  Bee, 


Vi=         =  17-9  ft.  per  sec., 

Y2  =  o7:H  =  15*7  ft.  per  sec. 

Inclination  of  the  vanes  at  inlet.  The  triangles  of  velocities 
for  the  three  radii  r,  ra  ,  r2  are  shown  in  Fig.  224.  For  example, 
at  radius  r,  ADC  is  the  triangle  of  velocities  at  inlet  and  ABC  the 


TURBINES  345 

triangle  of  velocities  at  outlet.     The  inclinations  of  the  vanes  at 
inlet  are  found  from 

0.1    ~ 

tan  <f>  =  og-7  _  14-4 »  f rom  which  <£  =  33°  30', 
tan *  =  17-98-21'5  and  *•  =  113° 50/' 

Q.1  * 

tan  02  =  TE^J —    A  •  -  5  from  which  <£2  =  137°  6'. 
JLO  /  —  Z4  O 

The  inclination  of  the  guide  blade  at  each  of  the  three  radii, 
fi     8-15 
=  2fr7> 
from  which  6  —  17°, 

tan  O-i  =  -j  ,-Q  and  01  =  24°  30', 

8'15 
tan«,=  j^an 

T/ie  inclination  of  the  vanes  at  exit. 


ten  0!=  f|t  =20°  48', 

Zl  o 

ten  «,=  |^=18'  22'. 


If  now  the  lower  tips  of  the  guide  blades  and  the  upper  tips 
of  the  wheel  vanes  are  made  radial  as  in  the  plan,  Fig.  221,  the 
inclination  of  the  guide  blade  will  have  to  vary  from  17  to 
27?,  degrees  or  else  there  will  be  loss  by  shock.  To  get  over  this 
difficulty  the  upper  edge  only  of  each  guide  blade  may  be  made 
radial,  the  lower  edge  of  the  guide  blade  and  the  upper  edge  of 
each  vane,  instead  of  being  radial,  being  made  parallel  to  the 
upper  edge  of  the  guide.  In  Fig.  225  let  r  and  E  be  the  radii 
of  the  inner  and  outer  crowns  of  the  wheel  and  also  of  the  guide 
blades.  Let  MN  be  the  plan  of  the  upper  edge  of  a  guide  blade 
and  let  DGr  be  the  plan  of  the  lower  edge,  DGr  being  parallel  to 
MN.  Then  as  the  water  runs  along  the  guide  at  D,  it  will  leave 
the  guide  in  a  direction  perpendicular  to  OD.  At  Gr  it  will  leave 
in  a  direction  HGr  perpendicular  to  OGr.  Now  suppose  the  guide 
at  the  edge  DGr  to  have  an  inclination  ft  to  the  plane  of  the  paper. 
If  then  a  section  of  the  guide  is  taken  by  a  vertical  plane  XX 
perpendicular  to  DGr,  the  elevation  of  the  tip  of  the  vane  on  this 
plane  will  be  AL,  inclined  at  ft  to  the  horizontal  line  AB,  and  AC 


846 


HYDRAULICS 


will  be  the  intersection  of  the  plane  XX  with  the  plane  tangent 
to  the  tip  of  the  vane. 

Now  suppose  DB  and  GrH  to  be  the  projections  on  the  plane 
of  the  paper  of  two  lines  lying  on  the  tangent  plane  AC  and 
perpendicular  to  CD  and  OGr  respectively.  Draw  EF  and  HK 
perpendicular  to  DE  and  GrH  respectively,  and  make  each  of 
them  equal  to  BC.  Then  the  angle  EDF  is  the  inclination  of  the 
stream  line  at  D  to  the  plane  of  the  paper,  and  the  angle  HGrK  is 
the  inclination  of  the  stream  line  at  Gr  to  the  plane  of  the  paper. 
These  should  be  equal  to  0  and  02  • 


k — 


p     t* 

Flew,  of  lower  edge  of  guide, 
bfadesl  of  upper  &dg&  of  rone, 

Fig.  225.     Plan  of  guide  blades  and  vanes  of  Parallel  Flow  Turbines. 

Let  y  be  the  perpendicular  distance  between  MN  and  DG. 
Let  the  angles  GOD  and  GOH  be  denoted  by  <£  an  da  respectively. 

Since  EF,  BC  and  HK  are  equal, 

EDtan0  =  #tan/3 (1), 

and  GH  tan  02  =  y  tan  j3 (2). 


But 

and 

Therefore 
and 

Again, 


-  cos  (a  +  </>), 


tan  0  =  cos  (a  +  <£)  tan  ft (3), 

tan  02  =  cos  a  tan  ft   (4). 

since- ^  ...(5). 


There  are  thus  three  equations  from  which  a,  <f>  and  ft  can  be 
determined. 

Let  x  and  y  be  the  coordinates  of  the  point  D,  0  being  the 
intersection  of  the  axes. 


TURBINES 


347 


Then 


and  from  (5) 


cos  (a  +  <£)  =  - , 


cosa  = 


Substituting  for  cos  (a  +  <£)  and  cos  a  and  the  known  values  of 
tan  0  and  tan  02  in  the  three  equations  (3  —  5),  three  equations  are 
obtained  with  a?,  y,  and  ft  as  the  unknowns. 

Solving  simultaneously 

x  =  1'14  feet, 

y  =  2'23  feet, 

and  tan  ft  =  0'67, 

from  which  ft  =  34°. 


Fig.  226. 


Fig.  228. 

The  length  of  the  guide  blade  is  thus  found,  and  the  constant 
slope  at  the  edge  DGr  so  that  the  stream  lines  at  D  and  Gr  shall 
have  the  correct  inclination. 

If  now  the  upper  edge  of  the  vane  is  just  below  DGr,  and  the 
tips  of  the  vane  at  D  and  Gr  are  made  as  in  Figs.  226 — 228,  <£  and 


348 


HYDRAULICS 


<£2  being  33°  30'  and  137°  6'  respectively,  the  water  will  move  on  to 
the  vane  without  shock. 

The  plane  of  the  lower  edge  of  the  vane  may  now  be  taken  as 
D'Gr',  Fig.  225,  and  the  circular  sections  DD',  PQ,  and  GrGT  at  the 
three  radii,  r,  ri,  and  r2  are  then  as  in  Figs.  226 — 228. 

198.     Regulation  of  the  flow  to  parallel  flow  turbines. 

To  regulate  the  flow  through  a  parallel  flow  turbine,  Fontaine 
placed  sluices  in  the  guide  passages,  as  in  Fig.  229,  connected  to 
a  ring  which  could  be  raised  or  lowered  by  three  vertical  rods 
having  nuts  at  the  upper  ends  fixed  to  toothed  pinions.  When 


Fig.  229.     Fontaine's  Sluices. 


Fig.  230.     Adjustable  guide  blades  for  Parallel  Flow  Turbine. 

the  sluices  required  adjustment,  the  nuts  were  revolved  together 
by  a  central  toothed  wheel  gearing  with  the  toothed  pinions 
carrying  the  nuts.  Fontaine  fixed  the  turbine  wheel  to  a  hollow 
shaft  which  was  carried  on  a  footstep  above  the  turbine.  In  some 
modern  parallel  flow  turbines  the  guide  blades  are  pivoted,  as  in 
Fig.  230,  so  that  the  flow  can  be  regulated.  The  wheel  may  be 
made  with  the  crowns  opening  outwards,  in  section,  similar  to 
the  Grirard  turbine  shown  in  Fig.  254,  so  that  the  axial  velocity 
with  which  the  water  leaves  the  wheel  may  be  small. 

The  axial  flow  turbine  is  well  adapted  to  low  falls  with  variable 
head,  and  may  be  made  in  several  compartments  as  in  Fig.  220. 
In  this  example,  only  the  inner  ring  is  provided  with  gates.  In 
dry  weather  flow  the  head  is  about  3  feet  and  the  gates  of  the 
inner  ring  can  be  almost  closed  as  the  outer  ring  will  give  the  full 


TURBINES  349 

power.  During  times  of  flood,  and  when  there  is  plenty  of  water, 
the  head  falls  to  2  feet,  and  the  sluices  of  the  inner  ring  are 
opened.  A  larger  supply  of  water  at  less  head  can  thus  be 
allowed  to  pass  through  the  wheel,  and  although,  due  to  the  shock 
in  the  guide  passages  of  the  inner  ring,  the  wheel  is  not  so  efficient, 
the  abundance  of  water  renders  this  unimportant. 

Example.  A  double  compartment  Jonval  turbine  has  an  outer  diameter  of 
12'  6"  and  an  inner  diameter  of  6  feet. 

The  radial  width  of  the  inner  compartment  is  V  9"  and  of  the  outer  compart- 
ment 1'  6".  Allowing  a  velocity  of  flow  of  3-25  ft.  per  second  and  supposing  the 
minimum  fall  is  1'  8",  and  the  number  of  revolutions  per  minute  14,  find  the  horse- 
power of  the  wheel  when  all  the  guide  passages  are  open,  and  find  what  portion  of 
the  inner  compartment  must  be  shut  off  so  that  the  horse-power  shall  be  the  same 
under  a  head  of  3  feet.  Efficiency  70  per  cent. 

Neglecting  the  thickness  of  the  blades, 

the  area  of  the  outer  compartment  =  ^  (12-52  -9'52)  =  52'6  sq.  feet. 

„      inner  „  =^  (9'52-62)  =  42-8  sq.  feet. 

Total  area  =  95  -4  sq.  feet. 

The  weight  of  water  passing  through  the  wheel  is 

W  =  95-4  x  62-4  x  3-25  Ibs.  per  sec. 

=  19,300  Ibs.  per  sec. 
and  the  horse-power  is 

19,300x1-66x0-7 

550 

Assuming  the  velocity  of  flow  constant  the  area  required  when  the  head 
is  3  feet  is 

40-8x33,000 

'~60x62-5x3x-7 
=  55-6  sq.  feet, 
or  the  outer  wheel  will  nearly  develop  the  horse-power  required. 

199.     Bernouilli's  equations  for  axial  flow  turbines. 

The  Bernouilli's  equations  for  an  axial  flow  turbine  can  be 
written  down  in  exactly  the  same  way  as  for  the  inward  and 
outward  flow  turbines,  page  335,  except  that  for  the  axial  flow 
turbine  there  is  no  centrifugal  head  impressed  on  the  water 
between  inlet  and  outlet. 


w      2g      w      2g 
from  which,  since  v  is  equal  to  Vi , 

p  + 


w  2g  2g     w  2g 

p     V2     Vv     <u?     Pl     YJ*     u? 
therefore     -  +  s  ----  +  cT  =      +  ~cT  +  t 

w     2g       g       2g     w       2g      2g       g 

Vv    ylV    p    u2    IL2    Pl   , 

and  ~=+o  --  —  -  ft/. 

g         g       w     2g      2g      w 


350  HYDRAULICS 

But  in  Fig.  220,      ^  +  ^  =  H0  +  ^  -  HP, 

and  -^  =  -^  +  ^i  • 

Therefore,  —  =  H  —  -^  -  H/  —  h/. 

99  ^9 

If  Ui  is  axial  and  equal  to  u,  as  in  Fig.  223, 


200.     Mixed  flow  turbines. 

By  a  modification  of  the  shape  of  the  vanes  of  an  inward  flow 
turbine,  the  mixed  flow  turbine  is  obtained.  In  the  inward  and 
outward  flow  turbine  the  water  only  acts  upon  the  wheel  while  it 
is  moving  in  a  radial  direction,  but  in  the  mixed  flow  turbine  the 
vanes  are  so  formed  that  the  water  acts  upon  them  also,  while 
flowing  axially. 


Fig.  231.     Mixed  Flow  Turbine. 

Fig.  231  shows  a  diagrammatic  section  through  the  wheel  of 
a  mixed  flow  turbine,  the  axis  of  which  is  vertical.     The  water 


TURBINES 


351 


enters  the  wheel  in  a  horizontal  direction  and  leaves  it  vertically, 
but  it  leaves  the  discharging  edge  of  the  vanes  in  different 
directions.  At  the  upper  part  B  it  leaves  the  vanes  nearly 
radially,  and  at  the  lower  part  A,  axially.  The  vanes  are  spoon- 
shaped,  as  shown  in  Fig.  232,  and  should  be  so  formed,  or  in  other 
words,  the  inclination  of  the  discharging  edge  should  so  vary, 
that  wherever  the  water  leaves  the  vanes  it  should  do  so  with  no 
component  in  a  direction  perpendicular  to  the  axis  of  the  turbine, 
i.e.  with  no  velocity  of  whirl.  The  regulation  of  the  supply  to 
•the  wheel  in  the  turbine  of  Fig.  231  is  effected  by  a  cylindrical 
sluice  or  speed  gate  between  the  fixed  guide  blades  and  the  wheel. 


Fig.  232.     Wheel  of  Mixed  Flow  Turbine. 

Fig.  233  shows  a  section  through  the  wheel  and  casing  of  a 
double  mixed  flow  turbine  having  adjustable  guide  blades  to 
regulate  the  flow.  Fig.  234  shows  a  half  longitudinal  section  of 
the  turbine,  and  Fig.  235  an  outside  elevation  of  the  guide  blade 
regulating  gear.  The  guide  blades  are  surrounded  by  a  large 


352 


HYDRAULICS 


vortex  chamber,  and  the  outer  tips  of  the  guide  blades  are  of 
variable  shapes,  Fig.  233,  so  as  to  diminish  shock  at  the  entrance 
to  the  guide  passages.  Bach  guide  blade  is  really  made  in  two 
parts,  one  of  which  is  made  to  revolve  about  the  centre  C,  while 
the  outer  tip  is  fixed.  The  moveable  parts  are  made  so  that  the 
flow  can  be  varied  from  zero  to  its  maximum  value.  It  will  be 


Fig.  233.     Section  through  wheel  and  guide  blades  of  Mixed  Flow  Turbine. 

noticed  that  the  mechanism  for  moving  the  guide  blades  is 
entirely  external  to  the  turbine,  and  is  consequently  out  of  the 
water.  A  further  special  feature  is  that  between  the  ring  R 
and  each  of  the  guide  blade  cranks  is  interposed  a  spiral  spring. 
In  the  event  of  a  solid  body  becoming  wedged  between  two  of 
the  guide  blades,  and  thus  locking  one  of  them,  the  adjustment  of 
the  other  guide  blades  is  not  interfered  with,  as  the  spring  con- 
nected to  the  locked  blade  by  its  elongation  will  allow  the  ring 
to  rotate. 

As  with  the  inward  and  outward  flow  turbine,  the  mixed 
flow  turbine  wheel  may  either  work  drowned,  or  exhaust  into  a 
"suction  tube." 


TURBINES 


353 


For  a  given  flow,  and  width  of  wheel,  the  axial  velocity 
with  which  the  water  finally  flows  away  from  the  wheel  being  the 
same  for  the  two  cases,  the  diameter  of  a  mixed  flow  turbine  can 
be  made  less  than  an  inward  flow  turbine.  As  shown  on  page  340, 
the  diameter  of  the  inward  flow  turbine  is  in  large  measure  fixed 


Fig.  234.     Half-longitudinal  section  of  Mixed  Flow  Turbine. 

by  the  diameter  of  the  exhaust  openings  of  the  wheel.  For  the 
same  axial  velocity,  and  the  same  total  flow,  whether  the  turbine 
is  an  inward  or  mixed  flow  turbine,  the  diameter  d  of  the  exhaust 
openings  must  be  about  equal.  The  external  diameter,  therefore, 
of  the  latter  will  be  much  smaller  than  for  the  former,  and  the 
L.  H.  23 


354 


HYDRAULICS 


general  dimensions  of  the  turbine  will  be  also  diminished.  For 
a  given  head  H,  the  velocity  v  of  the  inlet  edge  being  the  same  in 
the  two  cases,  the  mixed  flow  turbine  can  be  run  at  a  higher 
angular  velocity,  which  is  sometimes  an  advantage  in  driving 
dynamos. 


s  3 


TURBINES  355 

Form  of  the  vanes.  At  the  receiving  edge,  the  direction  of  the 
blade  is  found  in  the  same  way  as  for  an  inward  flow  turbine. 

ABC,  Fig.  236,  is  the  triangle  of  velocities,  and  BC  is  parallel 
to  the  tip  of  the  blade.  This  triangle  has  been  drawn  for  the  data 
of  the  turbine  shown  in  Figs.  233  —  235  ;  v  is  46'5  feet  per  second, 
and  from 


Y  =  33'5  feet  per  second. 
The  angle  <£  is  139  degrees. 


Triangle  of  Velocities 
at  receiving  edge 

Fig.  236. 

The  best  form  for  the  vane  at  the  discharge  is  somewhat 
difficult  to  determine,  as  the  exact  direction  of  flow  at  any  point 
on  the  discharging  edge  of  the  vane  is  not  easily  found.  The 
condition  to  be  satisfied  is  that  the  water  must  leave  the  wheel 
without  any  component  in  the  direction  of  motion. 

The  following  construction  gives  approximately  the  form  of 
the  vane. 

Make  a  section  through  the  wheel  as  in  Fig.  237.  The  outline 
of  the  discharge  edge  FGrH  is  shown.  This  edge  of  the  vane  is 
supposed  to  be  on  a  radial  plane,  and  the  plan  of  it  is,  therefore, 
a  radius  of  the  wheel,  and  upon  this  radius  the  section  is  taken. 

It  is  now  necessary  to  draw  the  form  of  the  stream  lines,  as 
they  would  be  approximately,  if  the  water  entered  the  wheel 
radially  and  flowed  out  axially,  the  vanes  being  removed. 

Divide  04,  Fig.  237,  at  the  inlet,  into  any  number  of  equal 
parts,  say  four,  and  subdivide  by  the  points  a,  6,  d,  e. 

Take  any  point  A,  not  far  from  c,  as  centre,  and  describe 
a  circle  MMi  touching  the  crowns  of  the  wheel  at  M  and  MI. 
Join  AM  and  AMi. 

Draw  a  flat  curve  Mi  MI  touching  the  lines  AM  and  AMi  in  M 
and  MI  respectively,  and  as  near  as  can  be  estimated,  perpendicular 

23—2 


356 


HYDRAULICS 


to  the  probable  stream  lines  through  a,  6,  d,  e,  which  can  be 
sketched  in  approximately  for  a  short  distance  from  04. 

Taking  this  curve  MMi  as  approximately  perpendicular  to  the 
stream  lines,  two  points/  and  g  near  the  centres  of  AM  and 
are  taken. 


Fig.  237. 

Let  the  radius  of  the  points  g  and  /  be  r  and  rL  respectively. 
If  any  point  Ci  on  MM!  is  now  taken  not  far  from  A,  the 
peripheral  area  of  Mci  is  nearly  27rrMci,  and  the  .peripheral  area 
of  MiCi  is  nearly  27rriM1Ci. 

On  the  assumption  that  the  mean  velocity  through  MiM  is 
constant,  the  flow  through  Mci  will  be  equal  to  that  through 
Mid,  when, 


TURBINES  357 

If,  therefore,  MMX  is  divided  at  the  point  cl  so  that 
MI  Ci  _  r 
McT~ri' 
the  point  Ci  will  approximately  be  on  the  stream  line  through  c. 

If  now  when  the  stream  line  ccl  is  carefully  drawn  in,  it  is 
perpendicular  to  MMi,  the  point  Ci  cannot  be  much  in  error. 

A  nearer  approximation  to  c,\  can  be  found  by  taking  new  values 
for  r  and  n,  obtained  by  moving  the  points  /  and  g  so  that  they 
more  nearly  coincide  with  the  centres  of  CiM  and  CiMi.  If  the 
two  curves  are  not  perpendicular,  the  curve  MMi  and  the  point  cl 
are  not  quite  correct,  and  new  values  of  r  and  n  will  have  to  be 
obtained  by  moving  the  points  /  and  g.  By  approximation  Ci  can 
be  thus  found  with  considerable  accuracy. 

By  drawing  other  circles  to  touch  the  crown  of  the  wheels,  the 
curves  M2M3,  M4M5  etc.  normal  to  the  stream  lines,  and  the  points 
c-2,  c3,  etc.  on  the  centre  stream  line,  can  be  obtained. 

The  curve  22,  therefore,  divides  the  stream  lines  into  equal 
parts. 

Proceeding  in  a  similar  manner,  the  curves  11  and  33  can  be 
obtained,  dividing  the  stream  lines  into  four  equal  parts,  and 
these  again  subdivided  by  the  curves  aa,  66,  dd,  and  ee,  which 
intersect  the  outlet  edge  of  the  vane  at  the  points  F,  Gr,  H  and  e 
respectively. 

To  determine  the  direction  of  the  tip  of  the  vane  at  points  on  the 
discharging  edge.  At  the  points  F,  Gr,  H,  the  directions  of  the 
stream  lines  are  known,  and  the  velocities  UF,  u&,  Uu  can  be  found, 
since  the  flows  through  01,  12,  etc.  are  equal,  and  therefore 

=  ~  . 

O7T 

Draw  a  tangent  FK  to  the  stream  line  at  F.  This  is  the  inter- 
section, with  the  plane  of  the  paper,  of  a  plane  perpendicular  to 
the  paper  and  tangent  to  the  stream  line  at  F. 

The  point  F  in  the  plane  of  FK  is  moving  perpendicular  to  the 
plane  of  the  paper  with  a  velocity  equal  to  w.R0,  <«>  being  the 
angular  velocity  of  the  wheel,  and  R0  the  radius  of  the  point  F. 

If  a  circle  be  struck  on  this  plane  with  K  as  centre,  this  circle 
may  be  taken  as  an  imaginary  discharge  circumference  of  an 
inward  flow  turbine,  the  velocity  v  of  which  is  wR0j  and  the  tip  of 
the  blade  is  to  have  such  an  inclination,  that  the  water  shall 
discharge  radially,  i.e.  along  FK,  with  a  velocity  UF  .  Turning  this 
circle  into  the  plane  of  the  paper  and  drawing  the  triangle  of 
velocities  FST,  the  inclination  «F  of  the  tip  of  the  blade  at  F  in 
the  plane  FK  is  obtained. 


OF  THE 

UNIVERSITY 


358 


HYDRAULICS 


At  Gr  the  stream  line  is  nearly  vertical,  but  o>R2  can  be  set  out 
in  the  plane  of  the  paper,  as  before,  perpendicular  to  UG  and  the 
inclination  «G,  on  this  plane,  is  found. 

At  H,  aH  is  found  in  tne  same  waF>  and  tne  direction  o±  the 
vane,  in  definite  planes,  at  other  points  on  its  outlet  edge,  can  be 

similarly  found. 


Fig.   238. 


Fig.  239. 

Sections  of  the  vane  by  planes  0Gb,  and  OiHd.  These  are 
shown  in  Figs.  238  and  239,  and  are  determined  as  follows. 

Imagine  a  vertical  plane  tangent  to  the  tip  of  the  vane  at 
inlet.  The  angle  this  plane  makes  with  the  tangent  to  the  wheel 
at  b  is  the  angle  </>,  Fig.  236.  Let  BC  of  the  same  figure  be  the 


TURBINES  359 

plan  of  a  horizontal  line  lying  in  this  plane,  and  BD  the  plan  of 
the  radius  of  the  wheel  at  b.     The  angle  between  these  lines  is  y. 
Let  ft  be  the  inclination  of  the  plane  OG6  to  the  horizontal. 
From  D,  Fig.  236,  set  out  DE,  inclined  to  BD  at  an  angle  /?, 
and  intersecting  AB  produced  in  E,  and  draw  BF  perpendicular 
toCB. 

Make  BF  equal  to  BE  and  join  OF. 

Now  set  out  a  triangle  BGrDi  having  BG-  equal  to  OF,  DXG 
equal  to  DE,  and  the  angle  BGrDi  a  right  angle.  In  the  figure 
Dj  and  D  happen  to  coincide. 

The  angle  BGD  is  the  angle  y1?  which  the  line  of  intersection  of 
the  plane  0Gb,  Fig.  237,  with  the  plane  tangent  to  the  inlet  tip  of 
the  vane,  makes  with  the  radius  06. 

In  Fig.  238  the  inclination  of  the  inlet  tip  of  the  blade  is  yx  as 
shown. 

To  determine  the  angle  a  at  the  outlet  edge,  resolve  UG,  Fig. 
237,  along  and  perpendicular  to  OGr,  UQG  being  the  component 
along  OG-. 

Draw  the  triangle  of  velocities  DEF,  Fig.  238. 
The  tangent  to  the  vane  at  D  is  parallel  to  FE. 
In  the  same  way,  the  section  on  the  plane  lid,  Fig.  237,  may  be 
determined;  the  inclination  at  the  inlet  is  y2,  Fig.  239. 

Mixed  flow  turbine  working  in  open  stream.  A  double  turbine 
working  in  open  stream  and  discharging  through  a  suction  tube 
is  shown  in  Fig.  243.  This  is  a  convenient  arrangement  for 
moderately  low  falls.  Turbines,  of  this  class,  of  1500  horse- 
power, having  four  wheels  on  the  same  shaft  and  working  under 
a  head  of  25  feet,  and  making  150  revolutions  per  minute,  have 
recently  been  installed  by  Messrs  Escher  Wyss  at  Wangen  an  der 
Aare  in  Switzerland. 

201.     Cone  turbine. 

Another  type  of  inward  flow  turbine,  which  is  partly  axial  and 
partly  radial,  is  shown  in  Fig.  241,  and  is  known  as  the  cone 
turbine.  It  has  been  designed  by  Messrs  Escher  Wyss  to  meet 
the  demand  for  a  turbine  that  can  be  adapted  to  variable  flows. 

The  example  shown  has  been  erected  at  Gusset  near  Lyons  and 
makes  120  revolutions  per  minute. 

The  wheel  is  divided  into  three  distinct  compartments,  the 
supply  of  water  being  regulated  by  three  cylindrical  sluices  S,  Si 
and  S2.  The  sluices  S  and  Si  are  each  moved  by  three  vertical 
spindles  such  as  A  and  AI  which  carry  racks  at  their  upper  ends. 
These  two  sluices  move  in  opposite  directions  and  thus  balance 
each  other.  The  sluice  S2  is  normally  out  of  action,  the  upper 


360 


HYDRAULICS 


compartment  being  closed.  At  low  heads  this  upper  compartment 
is  allowed  to  come  into  operation.  The  sluice  S2  carries  a  rack 
which  engages  with  a  pinion  P,  connected  to  the  vertical  shaft  T. 


Feet,5 


ZOFeeb 


Fig.  240. 

The  shaft  T  is  turned  by  hand  by  means  of  a  worm  and 
wheel  W.  When  it  is  desired  to  raise  the  sluice  S2,  it  is  revolved 
by  means  of  the  pinion  P  until  the  arms  F  come  between  collars 
D  and  E  on  the  spindles  carrying  the  sluice  Si,  and  the  sluice  S2 
then  rises  and  falls  with  Si .  The  pinion,  gearing  with  racks  on  A 
and  AI,  is  fixed  to  the  shaft  M,  which  is  rotated  by  the  rack  R 
gearing  with  the  bevel  pinion  Q.  The  rack  R,  is  rotated  by  two 
connecting  rods,  one  of  which  C  is  shown,  and  which  are  under 
the  control  of  the  hydraulic  governor  as  described  on  page  378. 
The  wheel  shaft  can  be  adjusted  by  nuts  working  on  the 
square-threaded  screw  shown,  and  is  carried  on  a  special  collar 
bearing  supported  by  the  bracket  B.  The  weight  of  the  shaft  is 
partly  balanced  by  the  water-pressure  piston  which  has  acting 
underneath  it  a  pressure  per  unit  area  equal  to  that  in  the  supply 
chamber.  The  dimensions  shown  are  in  millimetres. 


TURBINES 


361 


Fig.  241.     Cone  Turbine. 


362  HYDRAULICS 

202.  Effect  of  changing  the  direction  of  the  guide  blade, 
when  altering  the  flow  of  inward  flow  and  mixed  flow 
turbines. 

As  long  as  the  velocity  of  a  wheel  remains  constant,  the 
backward  head  impressed  on  the  water  by  the  wheel  is  the  same, 
and  the  pressure  head,  at  the  inlet  to  the  wheel,  will  remain 
practically  constant  as  the  guides  are  moved.  The  velocity  of 
flow  U,  through  the  guides,  will,  therefore,  remain  constant; 
but  as  the  angle  0,  which  the  guide  makes  with  the  tangent  to  the 
wheel,  diminishes  the  radial  component  u,  of  U,  diminishes. 


Fig.  242. 

Let  ABC,  Fig.  242,  be  the  triangle  of  velocities  for  full  opening, 
and  suppose  the  inclination  of  the  tip  of  the  blade  is  made  parallel 
to  BC.  On  turning  the  guides  into  the  dotted  position,  the  incli- 
nation being  <f>\ ,  the  triangle  of  velocities  is  ABCi ,  and  the  relative 
velocity  of  the  water  and  the  periphery  of  the  wheel  is  now  BCj 
which  is  inclined  to  the  vane,  and  there  is,  consequently,  loss  due 
to  shock. 

It  will  be  seen  that  in  the  dotted  position  the  tips  of  the  guide 
blades  are  some  distance  from  the  periphery  of  the  wheel  and  it  is 
probable  that  the  stream  lines  on  leaving  the  guide  blades  follow 
the  dotted  curves  SS,  and  if  so,  the  inclination  of  these  stream 
lines  to  the  tangent  to  the  wheel  will  be  actually  greater  than  <f>\, 
and  BCi  will  then  be  more  nearly  parallel  to  BC.  The  loss  may 
be  approximated  to  as  follows : 

As  the  water  enters  the  wheel  its  radial  component  will  remain 
unaltered,  but  its  direction  will  be  suddenly  changed  from  BCi  to 
BC,  and  its  magnitude  to  BC2;  CiC2  is  drawn  parallel  to  AB. 
A  velocity  equal  to  dC2  has  therefore  to  be  suddenly  impressed  on 
the  water. 

On  page  68  it  has  been  shown  that  on  certain  assumptions  the 


TURBINES 


363 


head  lost  when  the  velocity  of   a  stream  is  suddenly  changed 
from  Vi  to  v2  is 


that  is,  it  is  equal  to  the  head  due  to  the  relative  velocity  of 
Vi  and  v2. 

But  CiCa  is  the  relative  velocity  of  Bd  and  BC2,  and  therefore 
the  head  lost  at  inlet  may  be  taken  as 


2? 

k  being  a  coefficient  which  may  be  taken  as  approximately  unity. 

203.  Effect  of  diminishing  the  flow  through  turbines  on 
the  velocity  of  exit. 

If  water  leaves  a  wheel  radially  when  the  flow  is  a  maximum, 
it  will  not  do  so  for  any  other  flow. 

The  angle  of  the  tip  of  the  blade  at  exit  is  unalterable,  and  if 
u  and  UQ  are  the  radial  velocities  of  flow,  at  full  and  part  load 
respectively,  the  triangles  of  velocity  are  DBF  and  DEFi,  Fig.  243. 

For  part  flow,  the  velocity  with  which  the  water  leaves  the 
wheel  is  Ui.  If  this  is  greater  than  u,  and  the  wheel  is  drowned, 
or  the  exhaust  takes  place  into  the  air,  the  theoretical  hydraulic 
efficiency  is  less  than  for  full  load,  but  if  the  discharge  is  down  a 
suction  tube  the  velocity  with  which  the  water  leaves  the  tube  is 
less  than  for  full  flow  and  the  theoretical  hydraulic  efficiency  is 
greater  for  the  part  flow.  The  loss  of  head,  by  friction  in  the 
wheel  due  to  the  relative  velocity  of  the  water  and  the  vane, 
which  is  less  than  at  full  load,  should  also  be  diminished,  as  also, 
the  loss  of  head  by  friction  in  the  supply  and  exhaust  pipes. 
The  mechanical  losses  remain  practically  constant  at  all  loads. 


Fig.  243. 


Fig.  244. 


The  fact  that  the  efficiency  of  turbines  diminishes  at  part  loads 
must,  therefore,  in  large  measure  be  due  to  the  losses  by  shock 
being  increased  more  than  the  friction  losses  are  diminished. 

By  suitably  designing  the  vanes,  the  greatest  efficiency  of 
inward  flow  and  mixed  flow  turbines  can  be  obtained  at  some 
fraction  of  full  load. 


364 


HYDRAULICS 


204.     Regulation  of  the  flow  by  cylindrical  gates. 

When  the  speed  of  the  turbine  is  adjusted  by  a  gate  between 
the  guides  and  the  wheel,  and  the  flow  is  less  than  the  normal,  the 
velocity  U  with  which  the  water  leaves  the  guide  is  altered  in 
magnitude  but  not  in  direction. 

Let  ABC  be  the  triangle  of  velocities,  Fig.  244,  when  the  flow  is 
normal. 

Let  the  flow  be  diminished  until  the  velocity  with  which  the 
water  leaves  the  guides  is  U0,.  equal  to  AD. 

Then  BD  is  the  relative  velocity  of  U0  and  v,  and  UQ  is  the 
radial  velocity  of  flow  into  the  wheel. 

Draw  DK  parallel  to  AB.  Then  for  the  water  to  move  along 
the  vane  a  sudden  velocity  equal  to  KD  must  be  impressed  on 

&  (KD)2 
the  water,  and  there  is  a  head  lost  equal  to  — ^ . 

To  keep  the  velocity  U  more  nearly  constant  Mr  Swain  has 
introduced  the  gate  shown  in  Fig.  245.  The  gate  g  is  rigidly 
connected  to  the  guide  blades,  and  to  adjust  the  flow  the  guide 
blades  as  well  as  the  gate  are  moved.  The  effective  width  of  the 
guides  is  thereby  made  approximately  proportional  to  the  quantity 
of  flow,  and  the  velocity  U  remains  more  nearly  constant.  If  the 
gate  is  raised,  the  width  b  of  the  wheel  opening  will  be  greater 
than  bi  the  width  of  the  gate  opening,  and  the  radial  velocity  u0 


Fig.  245.     Swain  Gate. 


Fig.  246. 


TURBINES 


365 


into  the  wheel  will  consequently  be  less  than  the  radial  velocity  u 
from  the  guides.  If  U  is  assumed  constant  the  relative  velocity  of 
the  water  and  the  vane  will  suddenly  change  from  BC  to  BCi, 
Fig.  248.  Or  it  may  be  supposed  that  in  the  space  between  the 
guide  and  the  wheel  the  velocity  U  changes  from  AC  to  ACi. 

The  loss  of  head  will  now  be  — ~ — -— . 

205.  The  form  of  the  wheel  vanes  between  the  inlet  and 
outlet  of  turbines. 

The  form  of  the  vanes  between  inlet  and  outlet  of  turbines 
should  be  such,  that  there  is  no  sudden  change  in  the  relative 
velocity  of  the  water  and  the  wheel. 

Consider  the  case  of  an  inward  flow  turbine.  Having  given 
a  form  to  the  vane  and  fixed  the  width  between  the  crowns  of  the 
wheel  the  velocity  relative  to  the  wheel  at  any  radius  r  can  be 
found  as  follows. 

Take  any  circumferential  section  ef  at  radius  r,  Fig.  247.  Let 
b  be  the  effective  width  between  the  crowns,  and  d  the  effective 
width  ef  between  the  vanes,  and  let  q  be  the  flow  in  cubic  feet 
per  second  between  the  vanes  Ae  and  B/. 


Fig.  247.     Relative  velocity  of  the  water  and  the  vanes. 


Fig.  248. 


366  HYDRAULICS 

The  radial  velocity  through  e/is 

Ur=bd' 

Find  by  trial  a  point  0  near  the  centre  of  ef  such  that  a  circle 
drawn  with  0  as  centre  touches  the  vanes  at  M  and  MI. 

Suppose  the  vanes  near  e  and  /  to  be  struck  with  arcs  of  circles. 
Join  0  to  the  centres  of  these  circles  and  draw  a  curve  MCMi 
touching  the  radii  OM  and  OMi  at  M  and  Mi  respectively. 

Then  MCMi  will  be  practically  normal  to  the  stream  lines 
through  the  wheel.  The  centre  of  MCMi  may  not  exactly 
coincide  with  the  centre  of  ef,  but  a  second  trial  will  probably 
make  it  do  so. 

If  then,  b  is  the  effective  width  between  the  crowns  at  C, 


can  be  scaled  off  the  drawing  and  vr  calculated. 
The  curve  of  relative  velocities  for  varying  radii  can  then  be 
plotted  as  shown  in  the  figure. 


B 


Fig.  249. 

It  will  be  seen  that  in  this  case  the  curve  of  relative  velocities 
changes  fairly  suddenly  between  c  and  h.  By  trial,  the  vanes 
should  be  made  so  that  the  variation  of  velocity  is  as  uniform 
as  possible. 

If  the  vanes  could  be  made  involutes  of  a  circle  of  'radius  K0, 


TURBINES  367 

as  in  Fig.  249,  and  the  crowns  of  the  wheel  parallel,  the  relative 
velocity  of  the  wheel  and  the  water  would  remain  constant. 
This  form  of  vane  is  however  entirely  unsuitable  for  inward 
flow  turbines  and  could  only  be  used  in  very  special  cases  for 
outward  flow  turbines,  as  the  angles  <f>  and  0  which  the  involute 
makes  with  the  circumferences  at  A  and  B  are  not  independent, 
for  from  the  figure  it  is  seen  that, 

° 


i  •     .         O 

and  sin  9  —  ^7  » 

XV 

sin(9      R 

or  —.  —  -  =  —  . 

sin  9      r 

The  angle  0  must  clearly  always  be  greater  than  9. 

206.  The  limiting  head  for  a  single  stage  reaction 
turbine. 

Reaction  turbines  have  not  yet  been  made  to  work  under  heads 
higher  than  430  feet,  impulse  turbines  of  the  types  to  be  presently 
described  being  used  for  heads  greater  than  this  value. 

From  the  triangle  of  velocities  at  inlet  of  a  reaction  turbine, 
e.g.  Fig.  226,  it  is  seen  that  the  whirling  velocity  V  cannot  be 
greater  than 

v  +  ucot<j>. 

Assuming  the  smallest  value  for  9  to  be  30  degrees,  and  the 
maximum  value  for  u  to  be  0'25  \/2grH,  the  general  formula 

V»=eH 

g 

becomes,  for  the  limiting  case, 


If  v  is  assumed  to  have  a  limiting  value  of  100  feet  per  second, 
which  is  higher  than  generally  allowed  in  practice,  and  e  to 
be  0*8,  then  the  maximum  head  H  which  can  be  utilised  in  a  one 
stage  reaction  turbine,  is  given  by  the  equation 

25'6H-  346  >/H  =  10,000, 
from  which  H  =  530  feet. 

207.     Series  or  multiple  stage  reaction  turbines. 

Professor  Osborne  Reynolds  has  suggested  the  use  of  two 
or  more  turbines  in  series,  the  same  water  passing  through  them 
successively,  and  a  portion  of  the  head  being  utilised  in  each. 

For  parallel  flow  turbines,  Reynolds  proposed  that  the  wheels 


368 


HYDRAULICS 


and  fixed  blades  be  arranged  alternately  as  shown  in  Fig.  250*. 
This  arrangement,  although  not  used  in  water  turbines,  is  very 
largely  used  in  reaction  steam  turbines. 


Fig.  250. 


Figs.  251,  252.     Axial  Flow  Impulse  Turbine. 
Taken  from  Prof.  Reynolds'  Scientific  Papers,  Vol.  i. 


TURBINES  369 

208.     Impulse  turbines. 

Girard  turbine.     To  overcome  the  difficulty  of  diminution  of 
efficiency  with  diminution  of  flow, 
Girard  introduced,  about  1850,  the 
free  deviation  or  partial  admission 
turbine. 

Instead  of  the  water  being 
admitted  to  the  wheel  throughout 
the  whole  circumference  as  in  the 
reaction  turbines,  in  the  Grirard 
turbine  it  is  only  allowed  to  enter 
the  wheel  through  guide  passages 
in  two  diametrically  opposite 
quadrants  as  shown  in  Figs.  252— 
254.  In  the  first  two,  the  flow  is 
axial,  and  in  the  last  radial. 

In  Fig.  252  above  the  guide  crown  are  two  quadrant-shaped 
plates  or  gates  2  and  4,  which  are  made  to  rotate  about  a  vertical 
axis  by  means  of  a  toothed  wheel.  When  the  gates  are  over  the 
quadrants  2  and  4,  all  the  guide  passages  are  open,  and  by  turning 
the  gates  in  the  direction  of  the  arrow,  any  desired  number  of  the 
passages  can  be  closed.  In  Fig.  254  the  variation  of  flow  is- 
effected  by  means  of  a  cylindrical  quadrant-shaped  sluice,  which, 
as  in  the  previous  case,  can  be  made  to  close  any  desired  number 
of  the  guide  passages.  Several  other  types  of  regulators  for 
impulse  turbines  were  introduced  by  Girard  and  others. 

Fig.  253  shows  a  regulator  employed  by  Fontaine.  Above  the 
guide  blades,  and  fixed  at  the  opposite  ends  of  a  diameter  DD, 
are  two  indiarubber  bands,  the  other  ends  of  the  bands  being 
connected  to  two  conical  rollers.  The  conical  rollers  can  rotate 
on  journals,  formed  on  the  end  of  the  arms  which  are  connected 
to  the  toothed  wheel  TW.  A  pinion  P  gears  with  TW,  and  by 
rotating  the  spindle  carrying  the  pinion  P,  the  rollers  can  be  made 
to  unwrap,  or  wrap  up,  the  indiarubber  band,  thus  opening  or 
closing  the  guide  passages. 

As  the  Girard  turbine  is  not  kept  full  of  water,  the  whole  of 
the  available  head  is  converted  into  velocity  before  the  water 
enters  the  wheel,  and  the  turbine  is  a  pure  impulse  turbine. 

To  prevent  loss  of  head  by  broken  water  in  the  wheel,  the  air 
should  be  freely  admitted  to  the  buckets  as  shown  in  Figs.  252 
and  254. 

For  small  heads  the  wheel  must  be  horizontal  but  for  large 
heads  it  may  be  vertical. 

This  class   of   turbine  has  the  disadvantage   that  it  cannot 
L.  H.  24 


370 


HYDRAULICS 


run  drowned,  and  hence  must  always  be  placed  above  the  tail 
water.  For  low  and  variable  heads  the  full  head  cannot  therefore 
be  utilised,  for  if  the  wheel  is  to  be  clear  of  the  tail  water,  an 
amount  of  head  equal  to  half  the  width  of  the  wheel  must  of 
necessity  be  lost. 


Fig.  254.     Girard  Radial  flow  Impulse  Turbine. 

To  overcome  this  difficulty  Grirard  placed  the  wheel  in  an  air- 
tight tube,  Fig.  254,  the  lower  end  of  which  is  below  the  tail  water 
level,  and  into  which  air  is  pumped  by  a  small  auxiliary  air-pump, 
the  pressure  being  maintained  at  the  necessary  value  to  keep  the 
surface  of  the  water  in  the  tube  below  the  wheel. 


TURBINES  371 

Let  H  be  the  total  head  above  the  tail  water  level  of  the  supply 
water,  —  the  pressure  head  due  to  the  atmospheric  pressure,  H0 

the  distance  of  the  centre  of  the  wheel  below  the  surface  of  the 
supply  water,  and  Ji0  the  distance  of  the  surface  of  the  water  in 
the  tube  below  the  tail  water  level.  Then  the  air-pressure  in 
the  tube  must  be 

&+*., 

w 
and  the  head  causing  velocity  of  flow  into  the  wheel  is,  therefore, 


W  \W 

So  that  wherever  the  wheel  is  placed  in  the  tube  below  the  tail 
water  the  full  fall  H  is  utilised. 

This  system,  however,  has  not  found  favour  in  practice,  owing 
to  the  difficulty  of  preserving  the  pressure  in  the  tube. 

209.  The  form  of  the  vanes  for  impulse  turbines,  neg- 
lecting friction. 

The  receiving  tip  of  the  vane  should  be  parallel  to  the  relative 
velocity  Vr  of  the  water  and  the  edge  of  the  vane,  Fig.  255. 

At  exit  the  relative  velocity  vr,  Fig.  256,  neglecting  friction, 
must  be  equal  to  the  relative  velocity  Vr  at  inlet. 

If  the  angle  a  which  the  tip  of  the  vane  at  exit  makes  with 
the  direction  of  vl  is  known  the  triangle  of  velocities  can  be  drawn, 
by  setting  out  DB  equal  to  v:  and  EF  at  an  angle  a  with  it  and 
equal  to  Vr  .  Then  DF  is  the  velocity  with  which  the  water  leaves 
the  wheel. 

For  the  axial  flow  turbine  Vi  equals  v,  and  the  triangle  of 
velocities  at  exit  is  AGB,  Fig.  255. 

If  the  velocity  with  which  the  water  leaves  the  wheel  is  Ui, 
the  theoretical  hydraulic  efficiency  is 

H-SL 

E  ^      1     U*2 

H  IP 

and  is  independent  of  the  direction  of  Ui  . 

It  should  be  observed,  however,  that  in  the  radial  flow  turbine 
the  area  of  the  section  of  the  stream  by  the  circumference  of  the 
wheel,  for  a  given  flow,  will  depend  upon  the  radial  component  of 
Ui,  and  in  the  axial  flow  turbine  the  area  of  the  section  of  the 
stream  by  a  plane  perpendicular  to  the  axis  will  depend  upon  the 
axial  component  of  Ui  .  That  is,  in  each  case  the  area  will  depend 
upon  the  component  of  Ui  perpendicular  to  Vi  . 

24—2 


372 


HYDRAULICS 


Now  the  section  of  the  stream  must  not  fill  the  outlet  arearof 
the  wheel,  and  the  minimum  area  of  this  outlet  so  that  it  is  just 
not  filled  will  clearly  be  obtained  for  a  given  value  of  Ui  when  Ui 
is  perpendicular  to  Vi*,  or  is  radial  in  the  outward  flow  and  axial  in 
the  parallel  flow  turbine. 

For  the  parallel  flow  turbine  since  BC  and  BGr,  Fig.  255,  are 
equal,  Ui  is  clearly  perpendicular  to  vl  when 


and  the  inclinations  a  and  <f>  of  the  tips  of  the  vanes  are  equal. 


Figs.  255,  256. 


D 


Fig.  257. 

If  R  and  r  are  the  outer  and  inner  radii  of  the  radial  flow 
turbine  respectively, 

R 


It  is  often  stated  that  this  is  the  condition  for  maximum  efficiency  but  it  only 
is  so,  as  stated  above,  for  maximum  flow  for  the  given  machine.  The  efficiency 
>nly  depends  upon  the  magnitude  of  Ux  and  not  upon  its  direction. 


TURBINES  373 

For  Ui  to  be  radial 

Vr  =  Vi  sec  a 

v.K 
-~-  sec  a, 

y  y 

and  if  v  is  made  equal  to  -~  ,  Vr  from  Fig.  255  is  equal  to  77  sec  <£, 

Z  Z 

and  therefore, 


210.  Triangles  of  velocity  for  an  axial  flow  impulse  tur- 
bine considering  friction. 

The  velocity  with  which  the  water  leaves  the  guide  passages 
may  be  taken  as  from  0'94  to  0*97  \/2#H,  and  the  hydraulic  losses 
in  the  wheel  are  from  5  to  10  per  cent. 

If  the  angle  between  the  jet  and  the  direction  of  motion  of  the 
vane  is  taken  as  30  degrees,  and  U  is  assumed  as  0'95  \/20H,  and  v 
as  0'45\/2#H,  the  triangle  of  velocities  is  ABC,  Fig.  257. 

Taking  10  per  cent,  of  the  head  as  being  lost  in  the  wheel,  the 
relative  velocity  vr  at  exit  can  be  obtained  from  the  expression 


If  now  the  velocity  of  exit  Ui  be  taken  as  0*22  V20H,  and 
circles  with  A  and  B  as  centres,  and  Ui  and  vr  as  radii  be 
described,  intersecting  in  D,  ABD  the  triangle  of  velocities  at  exit 
is  obtained,  and  Ui  is  practically  axial  as  shown  in  the  figure. 
On  these  assumptions  the  best  velocity  for  the  rim  of  the  wheel  is 
therefore  '45  v2<?H  instead  of  "« 


The  head  lost  due  to  the  water  leaving  the  wheel  with  velocity 
u  is  '048H,  and  the  theoretical  hydraulic  efficiency  is  therefore 
95'2  per  cent. 

The  velocity  head  at  entrance  is  0'9025H  and,  therefore,  '097H 
has  been  lost  when  the  water  enters  the  wheel. 
The  efficiency,  neglecting  axle  friction,  will  be 
H  -  O'lH  -  0-048H  -  0-097H 

-TT 
=  76  per  cent,  nearly. 

211.     Impulse  turbine  for  high  heads. 

For  high  heads  Girard  introduced  a  form  of  impulse  turbine, 
of  which  the  turbine  shown  in  Figs.  258  and  259,  is  the  modern 
development. 

The  water  instead  of  being  delivered  through  guides  over  an 
arc  of  a  circle,  is  delivered  through  one  or  more  adjustable  nozzles. 


TURBINES  375 

In  the  example  shown,  the  wheel  has  a  mean  diameter  of  6'9  feet 
and  makes  500  revolutions  per  minute;  it  develops  1600  horse- 
power under  a  head  of  1935  feet. 

The  supply  pipe  is  of  steel  and  is  1'312  feet  diameter. 

The  form  of  the  orifices  has  been  developed  by  experience,  and 
is  such  that  there  is  no  sudden  change  in  the  form  of  the  liquid 
vein,  and  consequently  no  loss  due  to  shock. 

The  supply  of  water  to  the  wheel  is  regulated  by  the  sluices 
shown  in  Fig.  258,  which,  as  also  the  axles  carrying  the  same, 
are  external  to  the  orifices,  and  can  consequently  be  lubricated 
while  the  turbine  is  at  work.  The  sluices  are  under  the  control 
of  a  sensitive  governor  and  special  form  of  regulator. 

As  the  speed  of  the  turbine  tends  to  increase  the  regulator 
moves  over  a  bell  crank  lever  and  partially  closes  both  the  orifices, 
Any  decrease  in  speed  of  the  turbine  causes  the  reverse  action  to 
take  place. 

The  very  high  peripheral  speed  of  the  wheel,  205  feet  per 
second,  produces  a  high  stress  in  the  wheel  due  to  centrifugal 
forces.  Assuming  the  weight  of  a  bar  of  the  metal  of  which  the 
rim  is  made  one  square  inch  in  section  and  one  foot  long  as 
3'36  Ibs.,  the  stress  per  sq.  inch  in  the  hoop  surrounding  the 
wheel  is 

3'36.  i>2 
/=       9 
=  4400  Ibs.  per  sq.  inch. 

To  avoid  danger  of  fracture,  steel  laminated  hoops  are  shrunk 
on  to  the  periphery  of  the  wheel. 

The  crown  carrying  the  blades  is  made  independent  of  the  disc 
of  the  wheel,  so  that  it  may  be  replaced  when  the  blades  become 
worn,  without  an  entirely  new  wheel  being  provided. 

The  velocity  of  the  vanes  at  the  inner  periphery  is  171  feet  per 
second,  and  is,  therefore,  0'484  \/2gH. 

If  the  velocity  U  with  which  the  water  leaves  the  orifice  is 
taken  as  0'97  \/2gH,  and  the  angle  the  jet  makes  with  the  tangent 
to  the  wheel  is  30  degrees,  the  triangle  of  velocities  at  entrance  is 
ABC,  Fig.  260,  and  the  angle  <f>  is  53'5  degrees. 

The  velocity  vl  of  the  outer  edges  of  the  vanes  is  205  feet  per 
second,  and  assuming  there  is  a  loss  of  head  in  the  wheel,  equal  to 
6  per  cent,  of  H, 


2g      2g 
and  vr  =  123'5  ft.  per  second. 


376 


HYDRAULICS 


If  then  the  angle  a  is  30  degrees  the  triangle  of  velocities  at 
exit  is  DEF,  Fig.  261. 

The  velocity  with  which  the  water  leaves  the  wheel  is  then 
Ui  =  95  feet  per  sec.,  and  the  head  lost  by  this  velocity  is  140  feet 
or  -073H. 
A  v=m        J}___ 

5*T-8> 

A 


D 


Fig.  260. 


Fig.  261. 


The  head  lost  in  the  pipe  and  nozzle  is,  on  the  assumption 
made  above, 

H-(0'97)2H=0'06H, 

and  the  total  percentage  loss  of  head  is,  therefore, 


and  the  hydraulic  efficiency  is  80*7  per  cent. 


\\    >>  \1 

\\  \  V 

\    \              1 

\  \        \ 

//     I 

^^"^ 

m 

Fig.  262.     Pelton  Wheel. 


TURBINES 


377 


The  actual  efficiency  of  a  similar  turbine  at  full  load  was  found 
by  experiment  to  be  78  per  cent.,  which  allows  a  mechanical  loss 
of  2' 7  per  cent. 

212.     Pelton  wheel. 

A  form  of  impulse  turbine  now  very  largely  used  for  high  heads 
is  known  as  the  Pelton  wheel. 

A  number  of  cups,  as  shown  in  Figs.  262  and  266,  is  fixed  to  a 
wheel  which  is  generally  mounted  on  a  horizontal  axis.  The 
water  is  delivered  to  the  wheel  through  a  rectangular  shaped 
nozzle,  the  opening  of  which  is  generally  made  adjustable,  either 
by  means  of  a  hand  wheel  as  in  Fig.  262,  or  automatically  by  a 
regulator  as  in  Fig.  266. 

As  shown  on  page  276,  the  theoretical  efficiency  of  the  wheel  is 
unity  and  the  best  velocity  for  the  cups  is  one-half  the  velocity  of 
the  jet.  This  is  also  the  velocity  generally  given  to  the  cups 
in  actual  examples.  The  width  of  the  cups  is  from  2J  to 
4  times  the  thickness  of  the  jet,  and  the  width  of  the  jet  is  about 
twice  its  thickness. 

The  actual  efficiency  is  between  70  and  82  per  cent. 

Table  XXXVIII  gives  the  numbers  of  revolutions  per  minute, 
the  diameters  of  the  wheels  and  the  nett  head  at  the  nozzle  in 
a  number  of  examples. 

TABLE  XXXVIII. 
Particulars  of  some  actual  Pelton  wheels. 


Head 

in  feet 

Diameter 
of  wheel 
(two  wheels) 

Eevolutions 
per  minute 

V 

U 

H.  P. 

262 

39-4" 

375 

64-5 

129 

500 

*233 

7" 

2100 

64 

125 

5 

*197 

20" 

650 

56-5 

112 

10 

722 

39" 

650 

111 

215 

167 

382 

60" 

300 

79 

156 

144 

*289 

54" 

310 

73 

136 

400 

508 

90" 

200 

79 

180 

300 

*  Picard  Pictet  and  Co.,  the  remainder  by  Escher  Wyss  and  Co. 

213.     Oil  pressure  governor  or  regulator. 

The  modern  applications  of  turbines  to  the  driving  of  electrical 
machinery,  has  made  it  necessary  for  particular  attention  to  be 
paid  to  the  regulation  of  the  speed  of  the  turbines. 

The  methods  of  regulating  the  flow  by  cylindrical  speed  gates 
and  moveable  guide  blades  have  been  described  in  connection  with 


378 


HYDRAULICS 


various  turbines  but  the  means  adopted  for  moving  the  gates  and 
guides  have  not  been  discussed. 

Until  recent  years  some  form  of  differential  governor  was 
almost  entirely  used,  but  these  have  been  almost  completely 
superseded  by  hydraulic  and  oil  governors. 

Figs.  263  and  264  show  an  oil  governor,  as  constructed  by 
Messrs  Escher  Wyss  of  Zurich. 


Figs.  263,  264.     Oil  Pressure  Regulator  for  Turbines. 

A  piston  P  having  a  larger  diameter  at  one  end  than  at  the 
other,  and  fitted  with  leathers  I  and  Zi,  fits  into  a  double  cylinder 
Ci.  Oil  under  pressure  is  continuously  supplied  through  a  pipe  S 
into  the  annulus  A  between  the  pistons,  while  at  the  back  of  the 
large  piston  the  pressure  of  the  oil  is  determined  by  the  regulator. 


TURBINES 


379 


Suppose  the  regulator  to  be  in  a  definite  position,  the  space 
behind  the  large  piston  being  full  of  oil,  and  the 
turbine  running  at  its  normal  speed.  The  valve  V 
(an  enlarged  diagrammatic  section  is  shown  in 
Fig.  265)  will  be  in  such  a  position  that  oil  cannot 
enter  or  escape  from  the  large  cylinder,  and  the 
pressure  in  the  annular  ring  between  the  pistons 
will  keep  the  regulator  mechanism  locked. 

If  the  wheel  increases  in  speed,  due  to  a 
diminution  of  load,  the  balls  of  the  spring  loaded 
governor  Gr  move  outwards  and  the  sleeve  M 
rises.  For  the  moment,  the  point  D  on  the  lever 
MD  is  fixed,  and  the  lever  turns  about  D  as  a 
fulcrum,  and  thus  raises  the  valve  rod  NV.  This 
allows  oil  under  pressure  to  enter  the  large 
cylinder  and  the  piston  in  consequence  moves  to 


Fig.  265. 


the  right,  and  moves  the  turbine  gates  in  the  manner  described  later. 
As  the  piston  moves  to  the  right,  the  rod  R,  which  rests  on  the 
wedge  W  connected  to  the  piston,  falls,  and  the  point  D  of  the 
lever  MD  consequently  falls  and  brings  the  valve  Y  back  to  its 
original  position.  The  piston  P  thus  takes  up  a  new  position 
corresponding  to  the  required  gate  opening.  The  speed  of  the 
turbine  and  of  the  governor  is  a  little  higher  than  before,  the 
increase  in  speed  depending  upon  the  sensitiveness  of  the  governor. 
On  the  other  hand,  if  the  speed  of  the  wheel  diminishes,  the 
sleeve  M  and  also  the  valve  V  falls  and  the  oil  from  behind  the 
large  piston  escapes  through  the  exhaust  E,  the  piston  moving 
to  the  left.  The  wedge  W  then  lifts  the  fulcrum  D,  the  valve  V 
is  automatically  brought  to  its  central  position,  and  the  piston  P 
takes  up  a  new  position,  consistent  with  the  gate  opening  being 
sufficient  to  supply  the  necessary  water  required  by  the  wheel. 

A  hand  wheel  and  screw,  Fig.  264,  are  also  provided,  so  that 
the  gates  can  be  moved  by  hand  when  necessary. 

The  piston  P  is  connected  by  the  connecting  rod  BE  to  a  crank 
EF,  which  rotates  the  vertical  shaft  T.  A  double  crank  KK  is 
connected  by  the  two  coupling  rods  shown  to  a  rotating  toothed 
wheel  R,  Fig.  241,  turning  about  the  vertical  shaft  of  the  turbine, 
and  the  movement,  as  described  on  page  360,  causes  the  adjust- 
ment of  the  speed  gates. 

214.     Water  pressure  regulators  for  impulse  turbines. 
Fig.  266  shows  a  water  pressure  regulator  as  applied  to  regulate 
the  flow  to  a  Pel  ton  wheel. 

The  area  of  the  supply  nozzle  is  adjusted  by  a  beak  B  which 


380 


HYDRAULICS 


Figs.  266,  267.     Pelton  Wheel  and  Water  Pressure  Eegulator. 


TURBINES 


381 


rotates  about  the  centre  0.  The  pressure  of  the  water  in  the 
supply  pipe  acting  on  this  beak  tends  to  lift  it  and  thus  to  open 
the  orifice.  The  piston  P,  working  in  a  cylinder  C,  is  also  acted 
upon,  on  its  under  side,  by  the  pressure  of  the  water  in  the  supply 
pipe  and  is  connected  to  the  beak  by  the  connecting  rod  DE. 
The  area  of  the  piston  is  made  sufficiently  large  so  that  when  the 
top  of  the  piston  is  relieved  of  pressure  the  pull  on  the  connecting 
rod  is  sufficient  to  close  the  orifice. 

The  pipe  p  conveys  water  under  the  same  pressure,  to  the 
valve  Y,  which  may  be  similar  to  that  described  in  connection  with 
the  oil  pressure  governor,  Fig.  265. 

A  piston  rod  passes  through  the  top  of  the  cylinder,  and  carries 
a  nut,  which  screws  on  to  the  square  thread  cut  on  the  rod.  A 
lever  eg,  Fig.  268,  which  is  carried  on  the  fixed  fulcrum  e,  is  made 
to  move  with  the  piston.  A  link  /A  connects  ef  with  the  lever 
MN,  one  end  M  of  which  moves  with  the  governor  sleeve  and  the 
other  end  N  is  connected  to  the  valve  rod  NV.  The  valve  V  is 
shown  in  the  neutral  position. 


Fig.  268. 

Suppose  now  the  speed  of  the  turbine  to  increase.  The 
governor  sleeve  rises,  and  the  lever  MN  turns  about  the  fulcrum 
A  which  is  momentarily  at  rest.  The  valve  V  falls  and  opens  the 
top  of  the  cylinder  to  the  exhaust.  The  pressure  on  the  piston 
P  now  causes  it  to  rise,  and  closes  the  nozzle,  thus  diminishing 
the  supply  to  the  turbine.  As  the  piston  rises  it  lifts  again  the 
lever  MN  by  means  of  the  link  A/,  and  closes  the  valve  V.  A 
new  position  of  equilibrium  is  thus  reached.  If  the  speed  of  the 


382 


HYDRAULICS 


governor  decreases  the  governor  sleeve  falls,  the  valve  Y  rises, 
and  water  pressure  is  admitted  to  the  top  of  the  piston,  which  is 
then  in  equilibrium,  and  the  pressure  on  the  beak  B  causes  it  to 
move  upwards  and  thus  open  the  nozzle. 

Hydraulic  valve  for  water  regulator.  Instead  of  the  simple 
piston  valve  controlled  mechanically,  Messrs  Bscher  Wyss  use,  for 
high  heads,  a  hydraulic  double-piston  valve  Pp,  Fig.  269. 

This  piston  valve  has  a  small  bore  through  its  centre  by  means 
of  which  high  pressure  water  which  is  admitted  below  the  valve 
can  pass  to  the  top  of  the  large  piston  P.  Above  the  piston  is  a 
small  plug  valve  Y  which  is  opened  and  closed  by  the  governor. 


Fig.  269.     Hydraulic  valve  for  automatic  regulation. 

If  the  speed  of  the  governor  decreases,  the  valve  Y  is  opened, 
thus  allowing  water  to  escape  from  above  the  piston  valve,  and  the 
pressure  on  the  lower  piston  p  raises  the  valve.  Pressure  water  is 
thus  admitted  above  the  regulator  piston,  and  the  pressure  on  the 
beak  opens  the  nozzle.  As  the  governor  falls  the  valve  Y  closes, 
the  exhaust  is  throttled,  and  the  pressure  above  the  piston  P  rises. 
When  the  exhaust  through  Y  is  throttled  to  such  a  degree  that 
the  pressure  on  P  balances  the  pressure  on  the  under  face  of  the 
piston  p,  the  valve  is  in  equilibrium  and  the  regulator  piston  is 
locked. 


TURBINES  383 

If  the  speed  of  the  governor  increases,  the  valve  V  is  closed, 
and  the  excess  pressure  on  the  upper  face  of  the  piston  valve 
causes  it  to  descend,  thus  connecting  the  regulator  cylinder  to 
exhaust.  The  pressure  on  the  under  face  of  the  regulator  piston 
then  closes  the  nozzle. 

Filter.  Between  the  conduit  pipe  and  the  governor  valve  V, 
is  placed  a  filter,  Figs.  270  and  271,  to  remove  any  sand  or  grit 
contained  in  the  water. 

Within  the  cylinder,  on  a  hexagonal  frame,  is  stretched  a 
piece  of  canvas.  The  water  enters  the  cylinder  by  the  pipe  E,  and 
after  passing  through  the  canvas,  enters  the  central  perforated 
pipe  and  leaves  by  the  pipe  S. 


Figs.  270,  271.     Water  Filter  for  Impulse  Turbine  Regulator. 

To  clean  the  filter  while  at  work,  the  canvas  frame  is  revolved 
by  means  of  the  handle  shown,  and  the  cock  R  is  opened.  Each 
side  of  the  hexagonal  frame  is  brought  in  turn  opposite  the 
chamber  A,  and  water  flows  outwards  through  the  canvas  and 
through  the  cock  R,  carrying  away  any  dirt  that  may  have 
collected  outside  the  canvas. 

Auxiliary  valve  to  prevent  hammer  action.  When  the  pipe  line 
is  long  an  auxiliary  valve  is  frequently  fitted  on  the  pipe  near  to 
the  nozzle,  which  is  automatically  opened  by  means  of  a  cataract 
motion*  as  the  nozzle  closes,  and  when  the  movement  of  the  nozzle 
beak  is  finished,  the  valve"  slowly  closes  again. 

If  no  such  provision  is  made  a  rapid  closing  of  the  nozzle 
means  that  a  large  mass  of  water  must  have  its  momentum 
quickly  changed  and  very  large  pressures  may  be  set  up,  or  in 
other  words  hammer  action  is  produced,  which  may  cause  fracture 
of  the  pipe. 

When  there  is  an  abundant  supply  of  water,  the  auxiliary 
valve  is  connected  to  the  piston  rod  of  the  regulator  and  opened 
and  closed  as  the  piston  rod  moves,  the  valve  being  adjusted  so 
that  the  opening  increases  by  the  same  amount  that  the  area  of 
the  orifice  diminishes. 

*  See  Engineer,  Vol.  xc.,  p.  255. 


384  HYDRAULICS 

If  the  load  on  the  wheel  does  not  vary  through  a  large  range 
the  quantity  of  water  wasted  is  not  large. 

215.    Hammer  blow  in  a  long  turbine  supply  pipe. 

Let  L  be  the  length  of  the  pipe  and  d  its  diameter. 
The  weight  of  water  in  the  pipe  is 


Let  the  velocity  change  by  an  amount  dt?  in  time  dt.    Then  the 

Wdv 

rate  of  change  of  momentum  is  --TJ-,  and  on  a  cross  section  of 

got 

the  lower  end  of  the  column  of  water  in  the  pipe  a  force  P  must 
be  applied  equal  to  this. 

„  -o     w  w^d2  dv 

Therefore  P  =  7  --  —.  . 

4     g     ct 

Referring  to  Fig.  266,  let  b  be  the  depth  of  the  orifice  and  di  its 
width. 

Then,  if  r  is  the  distance  of  D  from  the  centre  about  which  the 
beak  turns,  and  n  is  the  distance  of  the  closing  edge  of  the  beak 
from  this  centre,  and  if  at  any  moment  the  velocity  of  the  piston 
is  v0  feet  per  second,  the  velocity  of  closing  of  the  beak  will  be 


In  any  small  element  of  time  dt  the  amount  by  which  the 
nozzle  will  close  is 


96=     a*. 

r 

Let  it  be  assumed  that  U,  the  velocity  of  flow  through  the 
nozzle,  remains  constant.  It  will  actually  vary,  due  to  the 
resistances  varying  with  the  velocity,  but  unless  the  pipe  is  very 
long  the  error  is  not  great  in  neglecting  the  variation.  If  then  v 
is  the  velocity  in  the  pipe  at  the  commencement  of  this  element  of 
time  and  v  -  Cv  at  the  end  of  it,  and  A  the  area  of  the  pipe, 

v.A=b.dl.U    ..............................  (1) 


and  (^-8^)A  =    6-a.^.U  ...............  (2). 

Subtracting  (2)  from  (1), 


or 

Ot 


TURBINES  385 

If  W  is  the  weight  of  water  in  the  pipe,  the  force  P  in  pounds 
that  will  have  to  be  applied  to  change  the  velocity  of  this  water 
by  dv  in  time  dt  is 


mi  -D  ri  Vo 

Therefore  ~r~  , 

g   T     j\. 

and  the  pressure   per  sq.  inch  produced  in  the  pipe  near  the 
nozzle  is 

W  n  dJJvo 
W  —  --  •  —  7-9-  . 
g  r     A2 

Suppose  the  nozzle  to  be  completely  closed  in  a  time  t  seconds, 
and  during  the  closing  the  piston  P  moves  with  simple  harmonic 
motion. 

Then  the  distance  moved  by  the  piston  to  close  the  nozzle  is 

br 

n' 

and  the  time  taken  to  move  this  distance  is  t  seconds. 
The  maximum  velocity  of  the  piston  is  then 


and  substituting  in  (3),  the  maximum  value  of  —  is,  therefore, 

c/c 

dv  _ 
ot~ 
and  the  maximum  pressure  per  square  inch  is 


i.U     ir.W.Q     •*    Wv 
Pm~      2gtA*      ~2g.t.A*~2t'  gA> 

where  Q  is  the  flow  in  cubic  feet  per  second  before  the  orifice 
began  to  close,  and  v  is  the  velocity  in  the  pipe. 

Example.  A  500  horse-power  Pelton  Wheel  of  75  per  cent,  efficiency,  and  working 
under  a  head  of  260  feet,  is  supplied  with  water  by  a  pipe  1000  feet  long  and 
2'  3"  diameter.  The  load  is  suddenly  taken  off,  and  the  time  taken  by  the 
regulator  to  close  the  nozzle  completely  is  5  seconds. 

On  the  assumption  that  the  nozzle  is  completely  closed  (1)  at  a  uniform  rate, 
and  (2)  with  simple  harmonic  motion,  and  that  no  relief  valve  is  provided, 
determine  the  pressure  produced  at  the  nozzle. 

The  quantity  of  water  delivered  to  the  wheel  per  second  when  working  at  full 
power  is 

500  x  33,000 


The  weight  of  water  in  the  pipe  is 

W  =  62-4xJ.  (2-25)2xlOOO 

=  250,000  Ibs. 
L.  H.  25 


386  HYDRAULICS 


21'7 
The  velocity  is  gTg^5'25  ft-  Per  sec- 

In  case  (1)  the  total  pressure  acting  on  the  lower  end  of  the  column  of  water  in 
pipe  is 


the  pipe  is 

p_  250,  000x5-25 

9  x  5 

=  8200  Ibs. 

The  pressure  per  sq.  inch  is 
ft  900 

p  =  Z±lL  =  14-5  Ibs.  per  sq.  inch. 


-272 


In  case  (2)  pm=^  -^^  =22-8  Ibs.  per  sq.  inch. 

&  i  .  g  .  A 


EXAMPLES. 

(1)  Find  the  theoretical  horse-power  of  an  overshot  water-wheel  22  feet 
diameter,  using  20,000,000  gallons  of  water  per  24  hours  under  a  total  head 
of  25  feet. 

(2)  An  overshot  water-wheel  has  a  diameter  of  24  feet,  and  makes  3'5 
revolutions  per  minute.     The  velocity  of  the  water  as  it  enters  the  buckets 
is  to  be  twice  that  of  the  wheel's  periphery. 

If  the  angle  which  the  water  makes  with  the  periphery  is  to  be  15 
degrees,  find  the  direction  of  the  tip  of  the  bucket,  and  the  relative  velocity 
of  the  water  and  the  bucket. 

(3)  The  sluice  of  an  overshot  water-wheel  12  feet  diameter  is  vertically 
above  the  centre  of  the  wheel.     The  surface  of  the  water  in  the  sluice 
channel  is  2  feet  6  inches  above  the  top  of  the  wheel  and  the  centre  of  the 
sluice  opening  is  8  inches  above  the  top  of  the  wheel.     The  velocity  of  the 
wheel  periphery  is  to  be  one-half  that  of  the  water  as  it  enters  the  buckets. 
Determine  the  number  of  rotations  of  the  wheel,  the  point  at  which  the 
water  enters  the  buckets,  and  the  direction  of  the  edge  of  the  bucket. 

(4)  An  overshot  wheel  25  feet  diameter  having  a  width  of  5  feet,  and 
depth  of  crowns  12  inches,  receives  450  cubic  feet  of  water  per  minute,  and 
makes  6  revolutions  per  minute.     There  are  64  buckets. 

The  water  enters  the  wheel  at  15  degrees  from  the  crown  of  the  wheel 
with  a  velocity  equal  to  twice  that  of  the  periphery,  and  at  an  angle  of  20 
degrees  with  the  tangent  to  the  wheel. 

Assuming  the  buckets  to  be  of  the  form  shown  in  Fig.  180,  the  length 
of  the  radial  portion  being  one -half  the  length  of  the  outer  face  of  the 
bucket,  find  how  much  water  enters  each  bucket,  and,  allowing  for  centri- 
fugal forces,  the  point  at  which  the  water  begins  to  leave  the  buckets. 

(5)  An  overshot  wheel  32  feet  diameter  has  shrouds  14  inches  deep, 
and  is  required  to  give  9  horse -power  when  making  5  revolutions  per  minute. 

Assuming  the  buckets  to  be  one-third  filled  with  water  and  of  the  same 
form  as  in  the  last  question,  find  the  width  of  the  wheel,  when  the  total 
fall  is  32  feet  and  the  efficiency  60  per  cent. 


TURBINES  387 

Assuming  the  velocity  of  the  water  in  the  penstock  to  be  If  times  that 
of  the  wheel's  periphery,  and  the  bottom  of  the  penstock  level  with  the  top 
of  the  wheel,  find  the  point  at  which  the  water  enters  the  wheel.  Find  also 
where  water  begins  to  discharge  from  the  buckets. 

(6)  A  radial  blade  impulse  wheel  of  the  same  width  as  the  channel  in 
which  it  runs,  is  15  feet  diameter.     The  depth  of  the  sluice  opening  is 
12  inches  and  the  head  above  the  centre  of  the  sluice  is  3  feet.     Assuming 
a  coefficient  of  velocity  of  0'8  and  that  the  edge  of  the  sluice  is  rounded  so 
that  there  is  no  contraction,  and  the  velocity  of  the  rim  of  the  wheel  is  0*4 
the  velocity  of  flow  through  the  sluice,  find  the  theoretical  efficiency  of 
the  wheel. 

(7)  An  overshot  wheel  has  a  supply  of  30  cubic  feet  per  second  on  a  fall 
of  24  feet. 

Determine  the  probable  horse-power  of  the  wheel,  and  a  suitable 
width  for  the  wheel. 

(8)  The  water  impinges  on  a  Poncelet  float  at  15°  with  the  tangent  to 
the  wheel,  and  the  velocity  of  the  water  is  double  that  of  the  wheel.    Find, 
by  construction,  the  proper  inclination  of  the  tip  of  the  float. 

(9)  In  a  Poncelet  wheel,  the  direction  of  the  jet  impinging  on  the  floats 
makes  an  angle  of  15°  with  the  tangent  to  the  circumference  and  the  tip  of 
the  floats  makes  an  angle  of  30°  with  the  same  tangent.     Supposing  the 
velocity  of  the  jet  to  be  20  feet  per  second,  find,  graphically  or  otherwise, 
(1)  the  proper  velocity  of  the  edge  of  the  wheel,  (2)  the  height  to  which  the 
water  will  rise  on  the  float  above  the  point  of  admission,  (3)  the  velocity 
and  direction  of  motion  of  the  water  leaving  the  float. 

(10)  Show  that  the  efficiency  of  a  simple  reaction  wheel  increases 
with  the  speed  when  frictional  resistances  are  neglected,  but  is  greatest 
at  a  finite  speed  when  they  are  taken  into  account. 

If  the  speed  of  the  orifices  be  that  due  to  the  head  (1)  find  the  efficiency, 
neglecting  friction ;  (2)  assuming  it  to  be  the  speed  of  maximum  efficiency, 
show  that  f  of  the  head  is  lost  by  friction,  and  ^  by  final  velocity  of  water. 

(11)  Explain  why,  in  a  vortex  turbine,  the  inner  ends  of  the  vanes  are 
inclined  backwards  instead  of  being  radial. 

(12)  An   inward  flow  turbine  wheel  has  radial  blades   at  the  outer 

periphery,  and  at  the  inner  periphery  the  blade  makes  an  angle  of  30°  with 

-p 

the  tangent.     The  total  head  is  70  feet  and  r=-  .    Find  the  velocity  of  the 
rim  of  the  wheel  if  the  water  discharges  radially.     Friction  neglected. 

(13)  The  inner  and  outer  diameters  of  an  inward  flow  turbine  wheel 
are  1  foot  and  2  feet  respectively.    The  water  enters  the  outer  circumference 
at  12°  with  the  tangent,  and  leaves  the  inner  circumference  radially.     The 
radial  velocity  of  flow  is  6  feet  at  both  circumferences.     The  wheel  makes 
3*6  revolutions  per  second.     Determine  the  angles  of  the  vanes  at  both 
circumferences,  and  the  theoretical  hydraulic  efficiency  of  the  turbine. 

(14)  Water  is  supplied  to  an  inward  flow  turbine  at  44  feet  per  second, 
and  at  10  degrees  to  the  tangent  to  the  wheel.     The  wheel  makes  200 

25—2 


388  HYDRAULICS 

revolutions  per  minute.  The  inlet  radius  is  1  foot  and  the  outer  radius 
2  feet.  The  radial  velocity  of  flow  through  the  wheel  is  constant. 

Find  the  inclination  of  the  vanes  at  inlet  and  outlet  of  the  wheel. 

Determine  the  ratio  of  the  kinetic  energy  of  the  water  entering  the 
wheel  per  pound  to  the  work  done  on  the  wheel  per  pound. 

(15)  The  supply  of  water  for  an  inward  flow  reaction  turbine  is  500 
cubic  feet  per  minute  and  the  available  head  is  40  feet.     The  vanes  are 
radial  at  the  inlet,   the   outer   radius   is  twice  the  inner,   the   constant 
velocity  of  flow  is  4  feet  per  second,   and  the  revolutions  are  350  per 
minute.     Find  the  velocity  of  the  wheel,  the  guide   and  vane   angles, 
the  inner  and  outer  diameters,  and  the  width  of  the  bucket  at  inlet  and 
outlet.    Lond.  Un.  1906. 

(16)  An  inward  flow  turbine  on  15  feet  fall  has  an  inlet  radius  of  1  foot 
and  an  outlet  radius  of  6  inches.     Water  enters  at  15°  with  the  tangent  to 
the  circumference  and  is  discharged  radially  with  a  velocity  of  3  feet  per 
second.     The  actual  velocity  of  water  at  inlet  is  22  feet  per  second.     The 
circumferential  velocity  of  the  inlet  surface  of  the  wheel  is  19^  feet  per 
second. 

Construct  the  inlet  and  outlet  angles  of  the  turbine  vanes. 
Determine  the  theoretical  hydraulic  efficiency  of  the  turbine. 
If  the  hydraulic  efficiency  of  the  turbine  is  assumed  80  per  cent,  find  the 
vane  angles. 

(17)  A  quantity  of  water  Q  cubic  feet  per  second  flows  through  a 
turbine,  and  the  initial  and  final  directions   and  velocities  are  known. 
Apply  the    principle    of    equality    of    angular    impulse    and    moment    of 
momentum  to  find  the  couple  exerted  on  the  turbine. 

(18)  The  wheel  of  an  inward  flow  turbine  has  a  peripheral  velocity  of 
50  feet  per  second.     The  velocity  of  whirl  of  the  incoming  water  is  40  feet 
per  second,  and  the  radial  velocity  of  flow  5  feet  per  second.     Determine 
the  vane  angle  at  inlet. 

Taking  the  flow  as  20  cubic  feet  per  second  and  the  total  losses  as 
20  per  cent,  of  the  available  energy,  determine  the  horse-power  of  the 
turbine,  and  the  head  H. 

If  5  per  cent,  of  the  head  is  lost  in  friction  in  the  supply  pipe,  and  the 
centre  of  the  turbine  is  15  feet  above  the  tail  race  level,  find  the  pressure 
head  at  the  inlet  circumference  of  the  wheel. 

(19)  An  inward  flow  turbine  is  required  to  give  200  horse-power  under 
a  head  of   100  feet  when  running  at  500  revolutions  per  minute.     The 
velocity  with  which  the  water  leaves  the  wheel  axially  may  be  taken  as 
10  feet  per  second,  and  the  wheel  is  to  have  a  double  outlet.    The  diameter 
of  the  outer  circumference  may  be  taken  as  If  times  the  inner.    Determine 
the  dimensions  of  the  turbine  and  the  angles  of  the  guide  blades  and 
vanes  of  the  turbine  wheel.     The  actual  efficiency  is  to  be  taken  as  75  per 
cent,  and  the  hydraulic  efficiency  as  80  per  cent. 

(20)  An  outward  flow  turbine  wheel  has  an  internal  diameter  of  5'249 
feet  and  an  external  diameter  of  6'25  feet.     The  head  above  the  turbine  is 
141-5  feet.     The  width  of  the  wheel  at  inlet  is  10  inches,  and  the  quantity 


TURBINES  389 

of  water  supplied  per  second  is  215  cubic  feet.  Assuming  the  hydraulic 
losses  are  20  per  cent.,  determine  the  angles  of  tips  of  the  vanes  so  that 
the  water  shall  leave  the  wheel  radially.  Determine  the  horse-power  of 
the  turbine  and  verify  the  work  done  per  pound  from  the  triangles  of 
velocities. 

(21)     The  total  head  available  for  an  inward-flow  turbine  is  100  feet. 

The  turbine  wheel  is  placed  15  feet  above  the  tail  water  level. 

When  the  flow  is  normal,  there  is  a  loss  of  head  in  the  supply  pipe  of 
3  per  cent,  of  the  head  ;  in  the  guide  passages  a  loss  of  5  per  cent.  ;  in  the 
wheel  9  per  cent.  ;  in  the  down  pipe  1  per  cent.  ;  and  the  velocity  of  flow 
from  the  wheel  and  in  the  supply  pipe,  and  also  from  the  down  pipe  is 
8  feet  per  second. 

The  diameter  of  the  inner  circumference  of  the  wheel  is  9£  inches  and 
of  the  outer  19  inches,  and  the  water  leaves  the  wheel  vanes  radially. 
The  wheel  has  radial  vanes  at  inlet. 

Determine  the  number  of  revolutions  of  the  wheel,  the  pressure  head  in 
the  eye  of  the  wheel,  the  pressure  head  at  the  circumference  to  the  wheel, 
the  pressure  head  at  the  entrance  to  the  guide  chamber,  and  the  velocity 
which  the  water  has  when  it  enters  the  wheel.  From  the  data  given 


9 

(22)  A  horizontal  inward  flow  turbine  has  an  internal  diameter  of 
5  feet  4  inches  and  an  external  diameter  of  7  feet.     The  crowns  of  the 
wheel  are  parallel  and  are  8  inches  apart.     The  difference  in  level  of  the 
head  and  tail  water  is  6  feet,  and  the  upper  crown  of  the  wheel  is  just  below 
the  tail  water  level.    Find  the  angle  the  guide  blade  makes  with  the  tangent 
to  the  wheel,  when  the  wheel  makes  32  revolutions  per  minute,  and  the 
flow  is  45  cubic  feet  per  second.     Neglecting  friction,  determine  the  vane 
angles,  the  horse-power  of  the  wheel  and  the  theoretical  hydraulic  efficiency. 

(23)  A  parallel  flow  turbine  has  a  mean  diameter  of  11  feet. 

The  number  of  revolutions  per  minute  is  15,  and  the  axial  velocity  of 
flow  is  3'5  feet  per  second.  The  velocity  of  the  water  along  the  tips  of  the 
guides  is  15  feet  per  second. 

Determine  the  inclination  of  the  guide  blades  and  the  vane  angles  that 
the  water  shall  enter  without  shock  and  leave  the  wheel  axially. 

Determine  the  work  done  per  pound  of  water  passing  through  the  wheel. 

(24)  The  diameter  of  the  inner  crown  of  a  parallel  flow  pressure  turbine 
is  5  feet  and  the  diameter  of  the  outer  crown  is  8  feet.     The  head  over  the 
wheel  is  12  feet.     The  number  of  revolutions  per  minute  is  52.     The  radial 
velocity  of  flow  through  the  wheel  is  4  feet  per  second. 

Assuming  a  hydraulic  efficiency  of  0'8,  determine  the  guide  blade  angles 
and  vane  angles  at  inlet  for  the  three  radii  2  feet  6  inches,  3  feet  3  inches 
and  4  feet. 

Assuming  the  depth  of  the  wheel  is  8  inches,  draw  suitable  sections  of 
the  vanes  at  the  three  radii. 

Find  also  the  width  of  the  guide  blade  in  plan,  if  the  upper  and  lower 
edges  are  parallel,  and  the  lower  edge  makes  a  constant  angle  with  the 


390  HYDRAULICS 

plane  of  the  wheel,  so  that  the  stream  lines  at  the  inner  and  the  outer 
crown  may  have  the  correct  inclinations. 

(25)  A  parallel  flow  impulse  turbine  works  under  a  head  of  64  feet. 
The  water  is  discharged  from  the  wheel  in   an  axial  direction  with  a 
velocity  due  to  a  head  of  4  feet.     The  circumferential  speed  of  the  wheel 
at  its  mean  diameter  is  40  feet  per  second. 

Neglecting  all  frictional  losses,  determine  the  mean  vane  and  guide 
angles.  Lond.  Un.  1905. 

(26)  An  outward  flow  impulse  turbine  has  an  inner  diameter  of  5  feet, 
an  external  diameter  of  6  feet  3  inches,  and  makes  450  revolutions  per 
minute. 

The  velocity  of  the  water  as  it  leaves  the  nozzles  is  double  the  velocity 
of  the  periphery  of  the  wheel,  and  the  direction  of  the  water  makes  an 
angle  of  30  degrees  with  the  circumference  of  the  wheel. 

Determine  the  vane  angle  at  inlet,  and  the  angle  of  the  vane  at  outlet  so 
that  the  water  shall  leave  the  wheel  radially. 

Find  the  theoretical  hydraulic  efficiency.  If  8  per  cent,  of  the  head 
available  at  the  nozzle  is  lost  in  the  wheel,  find  the  vane  angle  at  exit  that 
the  water  shall  leave  radially. 

What  is  now  the  hydraulic  efficiency  of  the  turbine  ? 

(27)  In  an  axial  flow  Girard  turbine,  let  V  be  the  velocity  due  to  the 
effective  head.     Suppose  the  water  issues  from  the  guide  blades  with  the 
velocity  (V95V,  and  is  discharged  axially  with  a  velocity  '12V.     Let  the 
velocity  of  the  receiving  and  discharging  edges  be  0*55  V. 

Find  the  angle  of  the  guide  blades,  receiving  and  discharging  angles  of 
wheel  vanes  and  hydraulic  efficiency  of  the  turbine. 

(28)  Water  is  supplied  to  an  axial  flow  impulse  turbine,  having  a  mean 
diameter  of  6  feet,  and  making  144  revolutions  per  minute,  under  a  head  of 
100  feet.    The  angle  of  the  guide  blade  at  entrance  is  30°,  and  the  angle  the 
vane  makes  with  the  direction  of  motion  at  exit  is  30°.     Eight  per  cent,  of 
the  head  is  lost  in  the  supply  pipe  and  guide.     Determine  the  relative 
velocity  of  water  and  wheel  at  entrance,  and  on  the  assumption  that  10  per 
cent,  of  the  total  head  is  lost  in  friction  and  shock  in  the  wheel,  determine 
the  velocity  with  which  the  water  leaves  the  wheel.     Find  the  hydraulic 
efficiency  of  the  turbine. 

(29)  The  guide  blades  of  an  inward  flow  turbine  are  inclined  at  30 
degrees,  and  the  velocity  U  along  the  tip  of  the  blade  is  60  feet  per  second. 
The  velocity  of  the  wheel  periphery  is  55  feet  per  second.    The  guide  blades 
are  turned  so  that  they  are  inclined  at  an  angle  of  15  degrees,  the  velocity 
U  remaining  constant.     Find  the  loss  of  head  due  to  shock  at  entrance. 

If  the  radius  of  the  inner  periphery  is  one-half  that  of  the  outer  and  the 
radial  velocity  through  the  wheel  is  constant  for  any  flow,  and  the  water 
left  the  wheel  radially  in  the  first  case,  find  the  direction  in  which  it  leaves 
in  the  second  case.  The  inlet  radius  is  twice  the  outlet  radius. 

(30)  The  supply  of  water  to  a  turbine  is  controlled  by  a  speed  gate 
between  the  guides  and  the  wheel.     If  when  the  gate  is  fully  open  the 
velocity  with  which  the  water  approaches  the  wheel  is  70  feet  per  second 


TURBINES  391 

and  it  makes  an  angle  of  15  degrees  with  the  tangent  to  the  wheel,  find 
the  loss  of  head  by  shock  when  the  gate  is  half  closed.  The  velocity  of 
the  inlet  periphery  of  the  wheel  is  75  feet  per  second. 

(31)  A  Pelton  wheel,  which  may  be  assumed  to  have  semi-cylindrical 
buckets,  is  2  feet  diameter.    The  available  pressure  at  the  nozzle  when  it 
is  closed  is  200  Ibs.  per  square  inch,  and  the  supply  when  the  nozzle  is 
open  is  100  cubic  feet  per  minute.     If  the  revolutions  are  600  per  minute, 
estimate  the  horse-power  of  the  wheel  and  its  efficiency. 

(32)  Show  that  the  efficiency  of  a  Pelton  wheel  is  a  maximum — 
neglecting  frictional  and  other  losses — when  the  velocity  of  the  cups  equals 
half  the  velocity  of  the  jet. 

25  cubic  feet  of  water  are  supplied  per  second  to  a  Pelton  wheel  through 
a  nozzle,  the  area  of  which  is  44  square  inches.  The  velocity  of  the  cups 
is  41  feet  per  second.  Determine  the  horse-power  of  the  wheel  assuming 
an  efficiency  of  75  per  cent. 


CHAPTER  X. 

PUMPS. 

Pumps  are  machines  driven  by  some  prime  mover,  and  used 
for  raising  fluids  from  a  lower  to  a  higher  level,  or  for  imparting 
energy  to  fluids.  For  example,  when  a  mine  has  to  be  drained 
the  water  may  be  simply  raised  from  the  mine  to  the  surface,  and 
work  done  upon  it  against  gravity.  Instead  of  simply  raising  the 
water  through  a  height  h,  the  same  pumps  might  be  used  to 
deliver  water  into  pipes,  the  pressure  in  which  is  wh  pounds  per 
square  foot. 

A  pump  can  either  be  a  suction  pump,  a  pressure  pump,  or 
both.  If  the  pump  is  placed  above  the  surface  of  the  water  in 
the  well  or  sump,  the  water  has  to  be  first  raised  by  suction; 
the  maximum  height  through  which  a  pump  can  draw  water, 
or  in  other  words  the  maximum  vertical  distance  the  pump  can 
be  placed  above  the  water  in  the  well,  is  theoretically  34  feet,  but 
practically  the  maximum  is  from  25  to  30  feet.  If  the  pump 
delivers  the  water  to  a  height  h  above  the  pump,  or  against  a 
pressure-head  h,  it  is  called  a  force  pump. 

216.     Centrifugal  and  turbine  pumps. 

Theoretically  any  reaction  turbine  could  be  made  to  work  as 
a  pump  by  rotating  the  wheel  in  the  opposite  direction  to  that  in 
which  it  rotates  as  a  turbine,  and  supplying  it  with  water  at  the 
circumference,  with  the  same  velocity,  but  in  the  inverse  direction 
to  that  at  which  it  was  discharged  when  acting  as  a  turbine.  Up 
to  the  present,  only  outward  flow  pumps  have  been  constructed, 
and,  as  will  be  shown  later,  difficulty  would  be  experienced  in 
starting  parallel  flow  or  inward  flow  pumps. 

Several  types  of  centrifugal  pumps  (outward  flow)  are  shown 
in  Figs.  272  to  276.. 

The  principal  difference  between  the  several  types  is  in  the 
form  of  the  casing  surrounding  the  wheel,  and  this  form  has  con- 
siderable influence  upon  the  efficiency  of  the  pump.  The  reason 


PUMPS 


393 


for  this  can  be  easily  seen  in  a  general  way  from  the  following 
consideration.  The  water  approaches  a  turbine  wheel  with  a 
high  velocity  and  in  a  direction  making  a  small  angle  with  the 
direction  of  motion  of  the  inlet  circumference  of  the  wheel,  and 


Fig.  272.     Diagram  of  Centrifugal  Pump. 

thus  it  has  a  large  velocity  of  whirl.  When  the  water  leaves  the 
wheel  its  velocity  is  small  and  the  velocity  of  whirl  should  be  zero. 
In  the  centrifugal  pump  these  conditions  are  entirely  reversed; 
the  water  enters  the  wheel  with  a  small  velocity,  and  leaves 


394 


HYDRAULICS 


it  with  a  high  velocity.  If  the  case  surrounding  the  wheel 
admits  of  this  velocity  being  diminished  gradually,  the  kinetic 
energy  of  the  water  is  converted  into  useful  work,  but  if  not,  it  is 
destroyed  by  eddy  motions  in  the  casing,  and  the  efficiency  of  the 
pump  is  accordingly  low. 

In  Fig.  272  a  circular  casing  surrounds  the  wheel,  and  prac- 
tically the  whole  of  the  kinetic  energy  of  the  water  when  it  leaves 
the  wheel  is  destroyed ;  the  efficiency  of  such  pumps  is  generally 
much  less  than  50  per  cent. 


Fig.  273.     Centrifugal  Pump  with  spiral  casing. 

The  casing  of  Fig.  273  is  made  of  spiral  form,  the  sectional 
area  increasing  uniformly  towards  the  discharge  pipe,  and  thus 
being  proportional  to  the  quantity  of  water  flowing  through  the 
section.  It  may  therefore  be  supposed  that  the  mean  velocity  of 
flow  through  any  section  is  nearly  constant,  and  that  the  stream 
lines  are  continuous. 

The  wheel  of  Fig.  274  is  surrounded  by  a  large  whirlpool 
chamber  in  which,  as  shown  later,  the  velocity  with  which  the 
water  rotates  round  the  wheel  gradually  diminishes,  and  the 
velocity  head  with  which  the  water  leaves  the  wheel  is  partly 
converted  into  pressure  head. 

The  same  result  is  achieved  in  the  pump  of  Figs.  275  and  276 


PUMPS  395 

by  allowing  the  water  as  it  leaves  the  wheel  to  enter  guide 
passages,  similar  to  those  used  in  a  turbine  to  direct  the  water 
to  the  wheel.  The  area  of  these  passages  gradually  increases 
and  a  considerable  portion  of  the  velocity  head  is  thus  converted 
into  pressure  head  and  is  available  for  lifting  water.' 

This  class  of  centrifugal  pump  is  known  as  the  turbine  pump. 


Fig.  274.     Diagram  of  Centrifugal  Pump  with  Whirlpool  Chamber. 

217.     Starting  centrifugal  or  turbine  pumps. 

A  centrifugal  pump  cannot  commence  delivery  unless  the  wheel, 
casing,  and  suction  pipe  are  full  of  water. 

If  the  pump  is  below  the  water  in  the  well  there  is  no  difficulty 
in  starting  as  the  casing  will  be  maintained  full  of  water. 

When  the  pump  is  above  the  water  in  the  well,  as  in  Fig.  272, 
a  non-return  valve  V  must  be  fitted  in  the  suction  pipe,  to  prevent 
the  pump  when  stopped  from  being  drained.  If  the  pump  becomes 
empty,  or  when  the  pump  is  first  set  to  work,  special  means  have 
to  be  provided  for  filling  the  pump  case.  In  large  pumps  the  air 
may  be  expelled  by  means  of  steam,  which  becomes  condensed  and 
the  water  rises  from  the  well,  or  they  should  be  provided  with 


396 


HYDRAULICS 


an  air-pump  or  ejector  as  an  auxiliary  to  the  pump.  Small  pumps 
can  generally  be  easily  filled  by  hand  through  a  pipe  such  as 
shown  at  P,  Fig.  276. 

With  some  classes  of  pumps,  if  the  pump  has  to  commence 
delivery  against  full  head,  a  stop  valve  on  the  rising  main, 
Fig.  296,  is  closed  until  the  pump  has  attained  the  speed  necessary 
to  commence  delivery*,  after  which  the  stop  valve  is  slowly 
opened. 


Fig.  275. 


Turbine  Pump. 


Fig.  276. 


It  will  be  seen  later  that,  under  special  circumstances,  other 
provisions  will  have  to  be  made  to  enable  the  pump  to  commence 
delivery. 

218.     Form  of  the  vanes  of  centrifugal  pumps. 

The  conditions  to  be  satisfied  by  the  vanes  of  a  centrifugal 
pump  are  exactly  the  same  as  for  a  turbine.  At  inlet  the  direction 
of  the  vane  should  be  parallel  to  the  direction  of  the  relative 
velocity  of  the  water  and  the  tip  of  the  vane,  and  the  velocity 
with  which  the  water  leaves  the  wheel,  relative  to  the  pump  case, 
is  the  vector  sum  of  the  velocity  of  the  tip  of  the  vane  and  the 
velocity  relative  to  the  vane. 

*  See  page  409. 


PUMPS  397 

Suppose  the  wheel  and  casing  of  Fig.  272  is  full  of  water,  and 
the  wheel  is  rotated  in  the  direction  of  the  arrow  with  such  a 
velocity  that  water  enters  the  wheel  in  a  known  direction  with  a 
velocity  U,  Fig.  277,  not  of  necessity  radial. 

Let  v  be  the  velocity  of  the  receiving  edge  of  the  vane  or  inlet 
circumference  of  the  wheel;  Vi  the  velocity  of  the  discharging 
circumference  of  the  wheel ;  Ui  the  absolute  velocity  of  the  water 
as  it  leaves  the  wheel ;  V  and  Yi  the  velocities  of  whirl  at  inlet 
and  outlet  respectively;  VP  and  vr  the  relative  velocities  of  the 
water  and  the  vane  at  inlet  and  outlet  respectively ;  u  and  HI  the 
radial  velocities  at  inlet  and  outlet  respectively. 

The  triangle  of  velocities  at  inlet  is  ACD,  Fig.  277,  and  if  the 
vane  at  A,  Fig.  272,  is  made  parallel  to  CD  the  water  will  enter 
the  wheel  without  shock. 


C  B 

velocities 


•idngle  oC  velocities  Triangle  of  velocities 

cub  inlet.  at  e^cct. 

Fig.  277.  Fig.  278. 

The  wheel  being  full  of  water,  there  is  continuity  of  flow,  and 
if  A  and  Ax  are  the  circumferential  areas  of  the  inner  and  outer 
circumferences,  the  radial  component  of  the  velocity  of  exit  at  the 
outer  circumference  is 

AM 


If  the  direction  of  the  tip  of  the  vane  at  the  outer  circum- 
ference is  known  the  triangle  of  velocities  at  exit,  Fig.  278,  can  be 
drawn  as  follows. 

Set  out  BGr  radially  and  equally  to  Ui,  and  BE  equal  to  VL 

Draw  GrF  parallel  to  BE  at  a  distance  from  BE  equal  to  Ui, 
and  EF  parallel  to  the  tip  of  the  vane  to  meet  GrF  in  F. 

Then  BF  is  the  vector  sum  of  BE  and  EF  and  is  the  velocity 
with  which  the  water  leaves  the  wheel  relative  to  the  fixed  casing. 

219.     Work  done  on  the  water  by  the  wheel. 

Let  R  and  r  be  the  radii  of  the  discharging  and  receiving 
circumferences  respectively. 

The  change  in  angular  momentum  of  the  water  as  it  passes 
through  the  wheel  is  ViR  +  ~Vr/g  per  pound  of  flow,  the  plus  sign 
being  used  when  V  is  in  the  opposite  direction  to  V1?  as  in 
Figs.  277  and  278. 


398  HYDRAULICS 

Neglecting  frictional  and  other  losses,  the  work  done  by  the 
wheel  on  the  water  per  pound  (see  page  275)  is 


, 


9      '   9 

If  U  is  radial,  as  in  Fig.  272,  Y  is  zero,  and  the  work  done  on 
the  water  by  the  wheel  is 


^  foot  Ibs.  per  Ib.  flow. 
y 

If  then  Ho,  Fig.  272,  is  the  total  height  through  which  the  water 
is  lifted  from  the  sump  or  well,  and  ud  is  the  velocity  with  which 
the  water  is  delivered  from  the  delivery  pipe,  the  work  done  on 
each  pound  of  water  is 

Ho+|2' 

and  therefore, 

—  =  H0  +  7; —  =  H. 

g  2g 

Let  (180°  -  <£)  be  the  angle  which  the  direction  of  the  vane  at 
exit  makes  with  the  direction  of  motion,  and  (180°  -  0)  the  angle 
which  the  vane  makes  with  the  direction  of  motion  at  inlet.  Then 
ACD  is  0  and  BEF  is  <£. 

In  the  triangle  HEF,  HE  =  HF  cot  <£,  and  therefore, 

YI  —  Vi  —  Ui  COt  <j>. 

The  theoretical  lift,  therefore,  is 

Ua      Vi  (vi  -  Uj.  cot  </>) 
JtiL  —  -tio  ~*~  n —  —  • 

If  Q  is  the  discharge  and  AI  the  peripheral  area  of  the  dis- 
charging circumference, 


V?  —  Vi  ~  COt  <£ 

and  H> ^ (1). 

9 
If,  therefore,  the  water  enters  the  wheel  without  shock  and  all 

T> 

resistances  are  neglected,  the  lift  is  independent  of  the  ratio  — ,  and 

depends  only  on  the  velocity  and  inclination  of  the  vane  at  the 
discharging  circumference. 

220.     Ratio  of  Vx  to  v,. 

As  in  the  case  of  the  turbine,  for  any  given  head  H,  Yi  and  v^ 
can  theoretically  have  any  values  consistent  with  the   product 


PUMPS 


399 


Vi^i  being  equal  to  gH,  the  ratio  of  Vi  to  Vi  simply  depending  upon 
the  magnitude  of  the  angle  <£. 

The  greater  the  angle  </>  is  made  the  less  the  velocity  Vi  of  the 
periphery  must  be  for  a  given  lift. 


\ 


if 


— */ 

Fig.  279. 


This  is  shown  at  once  by  equation  (1),  section  219,  and  is 
illustrated  in  Fig.  279.  The  angle  <f>  is  given  three  values, 
30  degrees,  90  degrees  and  150  degrees,  and  the  product  Yt;  and 
also  the  radial  velocity  of  flow  tti  are  kept  constant.  The  theo- 
retical head  and  also  the  discharge  for  the  three  cases  are  there- 
fore the  same.  The  diagrams  are  drawn  to  a  common  scale,  and  it 
can  therefore  be  seen  that  as  <j>  increases  vl  diminishes,  and  Ui 
the  velocity  with  which  the  water  leaves  the  wheel  increases. 

221.  The  kinetic  energy  of  the  water  at  exit  from  the 
wheel. 

Part  of  the  head  H  impressed  upon  the  water  by  the  wheel 
increases  the  pressure  head  between  the  inlet  and  outlet,  and  the 
remainder  appears  as  the  kinetic  energy  of  the  water  as  it  leaves 


400  HYDEAULICS 

U2 
the  wheel.     This  kinetic  energy  is  equal  to  -^- ,  and  can  only  be 

utilised  to  lift  the  water  if  the  velocity  can  be  gradually  diminished 
so  as  to  convert  velocity  head  into  pressure  head.  This  however 
is  not  very  easily  accomplished,  without  being  accompanied  by  a 
considerable  loss  by  eddy  motions.  If  it  be  assumed  that  the  same 

TJ  2 

proportion  of  the  head  7^-  in  all  cases  is  converted  into  useful 

work,  it  is  clear  that  the  greater  Ui,  the  greater  the  loss  by  eddy 
motions,  and  the  less  efficient  will  be  the  pump.  It  is  to  be  ex- 
pected, therefore,  that  the  less  the  angle  <f>,  the  greater  will  be 
the  efficiency,  and  experiment  shows  that  for  a  given  form  of 
casing,  the  efficiency  does  increase  as  <£  is  diminished. 

222.  Gross  lift  of  a  centrifugal  pump. 

Let  Jia  be  the  actual  height  through  which  water  is  lifted; 
Jis  the  head  lost  in  the  suction  pipe ;  ha  the  head  lost  in  the  delivery 
pipe ;  and  u&  the  velocity  of  flow  along  the  delivery  pipe. 

Any  other  losses  of  head  in  the  wheel  and  casing  are  incident 

Ud 

to  the  pump,  but  hs,  hd,  and  the  head  ^-  should  be  considered  as 

external  losses. 

The  gross  lift  of  a  pump  is  then 

h  =  ha  +  hs  +  hd  +  ~^- , 
and  this  is  always  less  than  H. 

223.  Efficiencies  of  a  centrifugal  pump. 
Manometric  efficiency.    The  ratio  fr  >  or 

g  .h 


e  = 


Q 


is  the  manometric  efficiency  of  the  pump  at  normal  discharge. 

The  reason  for  specifically  defining  e  as  the  manometric 
efficiency  at  normal  discharge  is  simply  that  the  theoretical  lift  H 
has  been  deduced  from  consideration  of  a  definite  discharge  Q, 
and  only  for  this  one  discharge  can  the  conditions  at  the  inlet  edge 
be  as  assumed. 

A  more  general  definition  is,  however,  generally  given  to  e,  and 
for  any  discharge  Q,  therefore,  the  manometric  efficiency  may 
be  taken  as  the  ratio  of  the  gross  lift  at  that  discharge  to  the 
theoretical  head 

^i2  —  ^i  A   c°t  *£ 

AI 

9 


CENTRIFUGAL    PUMPS  401 

This  manometric  efficiency  of  the  pump  must  not  be  confused 
with  the  efficiency  obtained  by  dividing  the  work  done  by  the 
pump,  by  the  energy  required  to  do  that  work,  as  the  latter  in 
many  pumps  is  zero,  when  the  former  has  its  maximum  value. 

Hydraulic  efficiency.  The  hydraulic  efficiency  of  a  pump  is 
the  ratio  of  the  work  done  on  the  pump  wheel  to  the  gross  work 
done  by  the  pump. 

Let  W  =  the  weight  of  water  lifted  per  second. 

Let  h  =  the  gross  head 

-  ha  +  h8  +  hd  +  ~-  . 

*g 

Let  E  =  the  work  done  on  the  pump  wheel  in  foot  pounds 
per  second. 

Let  eh  =  the  hydraulic  efficiency.     Then 

_W.h 
E    ' 

The  work  done  on  the  pump  wheel  is  less  than  the  work  done 
on  the  pump  shaft  by  the  belt  or  motor  which  drives  the  pump, 
by  an  amount  equal  to  the  energy  -lost  by  friction  at  the  bearings 
of  the  machine.  This  generally,  in  actual  machines,  can  be 
approximately  determined  by  running  the  machine  without  load. 

Actual  efficiency.  From  a  commercial  point  of  view,  what  is 
generally  required  is  the  ratio  of  the  useful  work  .done  by  the 
pump,  taking  it  as  a  whole,  to  the  work  done  on  the  pump  shaft. 

Let  E«  be  the  energy  given  to  the  pump  shaft  per  sec.  and. 
em  the  mechanical  efficiency  of  the  pump,  then 


and  the  actual  efficiency 


E. 

Gross  efficiency  of  the  pump.     The  gross  efficiency  of  the  pump 
itself,  including  mechanical  as  well  as  fluid  losses,  is 

W.h 


ea  = 


E. 


224.  Experimental  determination  of  the  efficiency  of  a 
centrifugal  pump. 

The  actual  and  gross  efficiencies  of  a  pump  can  be  determined 
directly  by  experiment,  but  the  hydraulic  efficiency  can  only  be 
determined  when  at  all  loads  the  mechanical  efficiency  of  the 
pump  is  known. 

To  find  the  actual  efficiency,  it  is  only  necessary  to  measure 
the  height  through  which  water  is  lifted,  the  quantity  of  water 

L.  H.  26 


402  HYDRAULICS 

discharged,  and  the  energy  E8  given  to  the  pump  shaft  in  unit 

time. 

A.  very  convenient  method  of  determining  E8  with  a  fair 
degree  of  accuracy  is  to  drive  the  pump  shaft  direct  by  an  electric 
motor,  the  efficiency  curve  *  for  which  at  varying  loads  is  known. 
A  better  method  is  to  use  some  form  of  transmission  dynamo- 
meter t. 

225.    Design  of  pump  to  give  a  discharge  Q. 

If  a  pump  is  required  to  give  a  discharge  Q  under  a  gross 
lift  h,  and  from  previous  experience  the  probable  manometric 
efficiency  e  at  this  discharge  is  known,  the  problem  of  determining 
suitable  dimensions  for  the  wheel  of  the  pump  is  not  difficult. 
The  difficulty  really  arises  in  giving  a  correct  value  to  e  and  in 
making  proper  allowance  for  leakage. 

This  difficulty  will  be  better  appreciated  after  the  losses  in 
various  kinds  of  pumps  have  been  considered.  It  will  then  be 
seen  that  e  depends  upon  the  angle  <£,  the  velocity  of  the  wheel, 
the  dimensions  of  the  wheel,  the  form  of  the  vanes  of  the  wheel, 
the  discharge  through  the  wheel,  and  upon  the  form  of  the  casing 
surrounding  the  wheel;  the  form  of  the  casing  being  just  as 
important,  or  more  important,  than  the  form  of  the  wheel  in 
determining  the  probable  value  of  e. 

Design  of  the  wheel  of  a  pump  for  a  given  discharge  under  a 
given  head.  If  a  pump  is  required  to  give  a  discharge  Q  under  an 
effective  head  ha,  the  gross  head  h  can  only  be  determined  if  hS) 

hd,  and  =^- ,  are  known. 

Any  suitable  value  can  be  given  to  the  velocity  Ud.  If  the 
pipes  are  long  it  should  not  be  much  greater  than  5  feet  per  second 
for  reasons  explained  in  the  chapter  on  pipes,  and  the  velocity  us 
in  the  suction  pipe  should  be  equal  to  or  less  than  ud.  The 
velocities  us  and  ud  having  been  settled,  the  losses  hs  and  hd  can  be 
approximated  to  and  the  gross  head  h  found.  In  the  suction  pipe, 
as  explained  on  page  395,  a  foot  valve  is  generally  fitted,  at  which, 
at  high  velocities,  a  loss  of  head  of  several  feet  may  occur. 
The  angle  <j>  is  generally  made  from  10  to  90  degrees.  Theoreti- 
cally, as  already  stated,  it  can  be  made  much  greater  than 
90  degrees,  but  the  efficiency  of  ordinary  centrifugal  pumps  might 
be  very  considerably  diminished  as  <£  is  increased. 

The  manometric  efficiency  e  varies  very  considerably;  with 
radial  blades  and  a  circular  casing,  the  efficiency  is  not  generally 

*  See  Electrical  Engineering,  Thomalen-Howe,  p.  195. 
t  See  paper  by  Stanton,  Proc.  Inst.  Mech.  Engs.,  1903. 


CENTRIFUGAL   PUMPS  403 

more  than  0'3  to  0'4.  With  a  vortex  chamber,  or  a  spiral  casing, 
and  the  vanes  at  inlet  inclined  so  that  the  tip  is  parallel  to  the 
relative  velocity  of  the  water  and  the  vane,  and  <£  not  greater  than 
90  degrees,  the  manometric  efficiency  e  is  from  0'5  to  0*75,  being 
greater  the  less  the  angle  <£,  and  with  properly  designed  guide 
blades  external  to  the  wheel,  e  is  from  0'6  to  '85. 

The  ratio  of  the  diameter  of  the  discharging  circumference  to 
the  inlet  circumference  is  somewhat  arbitrary  and  is  generally 
made  from  2  to  3.  Except  for  the  difficulty  of  starting  (see 
section  226),  the  ratio  might  with  advantage  be  made  much 
smaller,  as  by  so  doing  the  f  rictional  losses  might  be  considerably 
reduced.  The  radial  velocity  HI  may  be  taken  from  2  to  10  feet 
per  second. 

Having  given  suitable  values  to  u,  and  to  any  two  of  the  three 
quantities,  e,  v,  and  <£,  the  third  can  be  found  from  the  equation 
,  _  e  (vi  -  ViUi  cot  <fr) 

9 

The  internal  diameter  d  of  the  wheel  will  generally  be  settled  from 
consideration  of  the  velocity  of  flow  u2  into  the  wheel.  This  may 
be  taken  as  equal  to  or  about  equal  to  u}  but  in  special  cases 
it  may  be  larger  than  u. 

Then  if  the  water  is  admitted  to  the  wheel  at  both  sides,  as  in 
Fig.  273, 


from  which  d  can  be  calculated  when  u^  and  Q  are  known. 

Let  b  be  the  width  of  the  vane  at  inlet  and  B  at  outlet,  and  D 
the  diameter  of  the  outlet  circumference. 


Then  l>  > 

-rrdu 

and  B  =  -§- 

TTL)Ui 

If  the  water  moves  toward  the  vanes  at  inlet  radially,  the 
inclination  0  of  the  vane  that  there  shall  be  no  shock  is  such  that 


and  if  guide  blades  are  to  be  provided  external  to  the  wheel,  as  in 
Fig.  275,  the  inclination  a  of  the  tip  of  the  guide  blade  with  the 
direction  of  Vi  is  found  from 

Ui 

tan  a  =  —  . 

The  guide  passages  should  be  so  proportioned  that  the  velocity 
Ui  is  gradually  diminished  to  the  velocity  in  the  delivery  pipe. 

26—2 


404  HYDRAULICS 

Limiting  velocity  of  the  rim  of  the  wheel.  Quite  apart  from 
head  lost  by  friction  in  the  wheel  due  to  the  relative  motion  of 
the  water  and  the  wheel,  there  is  also  considerable  loss  of  energy 
external  to  the  wheel  due  to  the  relative  motion  of  the  water  and 
the  wheel.  Between  the  wheel  and  the  casing  there  is  in  most 
pumps  a  film  of  water,  and  between  this  film  and  the  wheel, 
frictional  forces  are  set  up  which  are  practically  proportional  to 
the  square  of  the  velocity  of  the  wheel  periphery  and  to  the  area 
of  the  wheel  crowns.  An  attempt  is  frequently  made  to  diminish 
this  loss  by  fixing  the  vanes  to  a  central  diaphragm  only,  the 
wheel  thus  being  without  crowns,  the  outer  casing  being  so 
formed  that  there  is  but  a  small  clearance  between  it  and  the 
outer  edges  of  the  vanes.  At  high  velocities  these  frictional  resist- 
ances may  be  considerable.  To  keep  them  small  the  surface  of 
the  wheel  crowns  and  vanes  must  be  made  smooth,  and  to  this 
end  many  high  speed  wheels  are  carefully  finished. 

Until  a  few  years  ago  the  peripheral  velocity  of  pump  wheels 
was  generally  less  than  50  feet  per  second,  and  the  best  velocity 
was  supposed  to  be  about  30  feet  per  second.  They  are  now,  how- 
ever, run  at  much  higher  speeds,  and  the  limiting  velocities  are 
fixed  from  consideration  of  the  stresses  in  the  wheel  due  to  centri- 
fugal forces.  Peripheral  velocities  of  nearly  200  feet  per  second 
are  now  frequently  used,  and  Rateau  has  constructed  small  pumps 
with  a  peripheral  velocity  of  250  feet  per  second  *. 

Example.  To  find  the  proportions  of  a  pump  with  radial  blades  at  outlet 
(i.e.  0  =  90°)  to  lift  10  cubic  feet  of  water  per  second  against  a  head  of  50  feet. 

Assume  there  are  two  suction  pipes  and  that  the  water  enters  the  wheel  from 
both  sides,  as  in  Fig.  273,  also  that  the  velocity  in  the  suction  and  delivery  pipes 
and  the  radial  velocity  through  the  wheel  are  6  feet  per  second,  and  the  manometric 
efficiency  is  75  per  cent. 

First  to  find  vlt 

Since  the  blades  are  radial,  -75  —  =  50, 

y 

from  which  1^  =  46  feet  per  sec. 

To  find  the  diameter  of  the  suction  pipes. 
The  discharge  is  10  cubic  feet  per  second,  therefore 

2.£d2.  6  =  10, 
from  which  d  =  l'03'  =  123". 

If  the  radius  E  of  the  external  circumference  be  taken  as  2r  and  r  is  taken  equal 
to  the  radius  of  the  suction  pipes,  then  R  =  12f",  and  the  number  of  revolutions 
per  second^will  be 


The  velocity  of  the  inner  edge  of  the  vane  is 
u  =  23  feet  per  sec. 

*  Engineer,  1902. 


CENTRIFUGAL    PUMPS  405 

The  inclination  of  the  vane  at  inlet  that  the  water  may  move  on  to  the  vane 
without  shock  is 


and  the  water  when  it  leaves  the  wheel  makes  an  angle  a  with  vl  such  that 


If  there  are  guide  blades  surrounding  the  wheel,  a  gives  the  inclination  of  these 
blades. 

The  width  of  the  wheel  at  discharge  is 


=  3£  inches  about. 
The  width  of  the  wheel  at  inlet  =  6^  inches. 

226.  The  centrifugal  head  impressed  on  the  water  by 
the  wheel. 

Head  against  which  a  pump  will  commence  to  discharge.  As 
shown  on  page  335,  the  centrifugal  head  impressed  on  the  water  as 
it  passes  through  the  wheel  is 

,  _vl_rf 
~2<7     2g> 

but  this  is  not  the  lift  of  the  pump.  Theoretically  it  is  the  head 
which  will  be  impressed  on  the  water  when  there  is  no  flow 
through  the  wheel,  and  is  accordingly  the  difference  between  the 
pressure  at  inlet  and  outlet  when  the  pump  is  first  set  in  motion  ; 
or  it  is  the  statical  head  which  the  pump  will  maintain  when 

running  at  its  normal  speed.     If  this  is  less  than  -    -  ,  the  pump 

y 

theoretically  cannot  start  lifting  against  its  full  head  without 
being  speeded  up  above  its  normal  velocity. 

The  centrifugal  head  is,  however,  always  greater  than 


as  the  water  in  the  eye  of  the  wheel  and  in  the  casing  surrounding 
the  wheel  is  made  to  rotate  by  friction. 

For  a  pump  having  a  wheel  seven  inches  diameter  surrounded 
by  a  circular  casing  20  inches  diameter,  Stanton*  found  that,  when 
the  discharge  was  zero  and  the  vanes  were  radial  at  exit,  hc  was 

I'OW  ,  ,    .  T12t;2 

—  ~  —  ,  and  with  curved  vanes,  <£  being  30  degrees,  hc  was  —  ~  —  . 

^9  ^9 

For  a  pump  with  a  spiral  case  surrounding  the  wheel,  the 
centrifugal  head  hc  when  there  is  no  discharge,  cannot  be  much 

2 

greater  than  ~  ,  as  the  water  surrounding  the  wheel  is  prevented 

*9 

from  rotating  by  the  casing  being  brought  near  to  the  wheel  at 
one  point. 

*  Proceedings  Inst.  M.  E.,  1903. 


406  HYDRAULICS 

Parsons  found  for  a  pump  having  a  wheel  14  inches  diameter 
with  radial  vanes  at  outlet,  and  running  at  300  revolutions  per 

minute,  that  the  head  maintained  without  discharge  was  —  ~  —  , 
and  with  an  Appold*  wheel  running  at  320  revolutions  per  minute 
the  statical  head  was  —  ^  —  .  For  a  pump,  with  spiral  casing, 

having  a  rotor  1*54  feet  diameter,  the  least  velocity  at  which 
it  commenced  to  discharge  against  a  head  of  14*67  feet  was 


392  revolutions  per  minute,  and  thus  hc  was    ~   *  >  an(i  the  least 

velocity  against   a  head   of  17*4  feet  was  424  revolutions   per 

0*951)  2 
minute  or  hc  was  again  —  ~  —  ~  •     For  a  pump  with  circular  casing 

larger  than  the  wheel,  hc  was  —  x  —  -  .     For  a  pump  having  guide 

passages  surrounding  the  wheel,  and  outside  the  guide  passages 
a  circular  chamber  as  in  Fig.  275,  the  centrifugal  head  may  also 

2 

be  larger  than  <£-;  the  mean  actual  value  for  this  pump   was 

9  2 
found  to  be  1*087  ^-. 

Stanton  found,  when  the  seven  inches  diameter  wheels  mentioned 
above  discharged  into  guide  passages  surrounded  by  a  circular 

chamber  20  inches  diameter,  that  hc  was  —  ^  —  -  when  the  vanes  of 

1*39?;  2 
the  wheel  were  radial,  and  —  ^  —  -  when  <f>  was  30  degrees. 

That  the  centrifugal  head  when  the  wheel  has  radial  vanes  is 
likely  to  be  greater  than  when  the  vanes  of  the  wheel  are  set  back 
is  to  be  seen  by  a  consideration  of  the  manner  in  which  the  water 
in  the  chamber  outside  the  guide  passages  is  probably  set  in 
motion,  Fig.  280.  Since  there  is  no  discharge,  this  rotation  cannot 
be  caused  by  the  water  passing  through  the  pump,  but  must  be 
due  to  internal  motions  set  up  in  the  wheel  and  casing.  The 
water  in  the  guide  chamber  cannot  obviously  rotate  about  the 
axis  0,  but  there  is  a  tendency  for  it  to  do  so,  and  consequently 
stream  line  motions,  as  shown  in  the  figure,  are  probably  set 
up.  The  layer  of  water  nearest  the  outer  circumference  of  the 
wheel  will  no  doubt  be  dragged  along  by  friction  in  the  direction 
shown  by  the  arrow,  and  water  will  flow  from  the  outer  casing  to 
take  its  place  ;  the  stream  lines  will  give  motion  to  the  water  in 
the  outer  casing. 

*  See  page  415. 


CENTRIFUGAL    PUMPS 


407 


When  the  vanes  in  the  wheel  are  radial  and  as  long  as  a  vane  is 
moving  between  any  two  guide  vanes,  the  straight  vane  prevents 
the  friction  between  the  water  outside  the  wheel  and  that  inside, 
from  dragging  the  water  backwards  along  the  vane,  but  when  the 
vane  is  set  back  and  the  angle  <£  is  greater  than  90  degrees,  there 
will  be  a  tendency  for  the  water  in  the  wheel  to  move  backwards 
while  that  in  the  guide  chamber  moves  forward,  and  consequently 
the  velocity  of  the  stream  lines  in  the  casing  will  be  less  in  the 
latter  case  than  in  the  former.  In  either  case,  the  general 
direction  of  flow  of  the  stream  lines,  in  the  guide  chamber,  will 
be  in  the  direction  of  rotation  of  the  wheel,  but  due  to  friction 
and  eddy  motions,  even  with  radial  vanes,  the  velocity  of  the  stream 


Fig.  280. 

lines  will  be  less  than  the  velocity  t\  of  the  periphery  of  the  wheel. 
Just  outside  the  guide  chambers  the  velocity  of  rotation  will  be 
less  than  Vi.  In  the  outer  chamber  it  is  to  be  expected  that  the 
water  will  rotate  as  in  a  free  vortex,  or  its  velocity  of  whirl  will 
be  inversely  proportional  to  the  distance  from  the  centre  of  the 
rotor,  or  will  rotate  in  some  manner  approximating  to  this. 

The  head  which  a  pump,  with  a  vortex  chamber,  will  theoreti- 
cally maintain  when  the  discharge  is  zero.  In  this  case  it  is 
probable  that  as  the  discharge  approaches  zero,  in  addition  to  the 
water  in  the  wheel  rotating,  the  water  in  the  vortex  chamber  will 
also  rotate  because  of  friction. 


408  HYDRAULICS 

The  centrifugal  head  due  to  the  water  in  the  wheel  is 

2g~2g' 

q  ,..  2 

If  R  =  2r,  this  becomes  7  <r-  • 

The  centrifugal  head  due  to   the  water  in   the   chamber  is, 
Fig.  281, 


TO  and  VQ  being  the  radius  and  tangential  velocity  respectively  of 
any  ring  of  water  of  thickness  dr. 


Fig.  281. 

If  it  be  assumed  that  v0r0  is  a  constant,  the  centrifugal  head 
due  to  the  vortex  chamber  is 


The  total  centrifugal  head  is  then 

z,       v* 
c  =  2 — 

If  rw  is  2r  and  E,w  is  2rw, 


_vf_v^     v^/J.         1 

IVr:    r^.  /-».  ~T~      ^-v  I  «    ~~"     -w^ 


The  conditions  here  assumed,  however,  give  he  too  high.     In 
Stanton's  experiments  hc  was  only     ^  Vl  .     Decouer  from  experi- 


CENTRIFUGAL   PUMPS  409 

ments  on  a  small  pump  with  a  vortex  chamber,  the  diameter  being 
about  twice  the  diameter  of  the  wheel,  found  hc  to  be     ~  l  . 

Let  it  be  assumed  that  hc  is  -~-  in  any  pump,  and  that  the  lift 

of  the  pump  when  working  normally  is 

e  (i?i2  -  ViUi  cot 


Then  if  h  is  greater  than  -^  l  ,  the  pump  will  not  commence  to 

t/ 

discharge  unless  speeded  up  to  some  velocity  vz  such  that 

e  (vi2  —  Vi  HI  cot  <£) 


After  the  discharge  has  been  commenced,  however,  the  speed 
may  be  diminished,  and  the  pump  will  continue  to  deliver  against 
the  given  head*. 

For  any  given  values  of  m  and  e  the  velocity  v2  at  which  delivery 
commences  decreases  with  the  angle  <£.  If  <£  is  90  or  greater  than 
90  degrees,  and  m  is  unity,  the  pump  will  only  commence  to 
discharge  against  the  normal  head  when  the  velocity  is  v^  ,  if  the 
manometric  efficiency  e  is  less  than  0'5.  If  <£  is  30  degrees  and  m 
is  unity,  v%  is  equal  to  Vi  when  e  is  0'6,  but  if  <£  is  150  degrees  vz 
is  equal  to  vl  when  e  is  0'428. 

Nearly  all  actual  pumps  are  run  at  such  a  speed  that  the 
centrifugal  head  at  that  speed  is  greater  than  the  actual  head 
against  which  the  pump  works,  so  that  there  is  never  any 
difficulty  in  starting  the  pump.  This  is  accounted  for  (1)  by  the 
low  manometric  efficiencies  of  actual  pumps,  (2)  by  the  angle  <j> 
never  being  greater  than  90  degrees,  and  (3)  by  the  wheels  being 
surrounded  by  casings  which  allow  the  centrifugal  head  to  be 

greater  than  ^-  . 

It  should  be  observed  that  it  does  not  follow,  because  in  many 
cases  the  manometric  efficiency  is  small,  the  actual  efficiency  of 
the  pump  is  of  necessity  low.  (See  Fig.  286.) 

227.  Head-velocity  curve  of  a  centrifugal  pump  at  zero 
discharge. 

For  any  centrifugal  pump  a  curve  showing  the  head  against 
which  it  will  start  pumping  at  any  given  speed  can  easily  be 
determined  as  follows. 

On  the  delivery  pipe  fit  a  pressure  gauge,   and  at  the   top 

*  See  pages  411  and  419. 


410 


HYDRAULICS 


of  the  suction  pipe  a  vacuum  gauge.  Start  the  pump  with 
the  delivery  valve  closed,  and  observe  the  pressure  on  the  two 
gauges  for  various  speeds  of  the  pump.  Let  p  be  the  absolute 
pressure  per  sq.  foot  in  the  delivery  pipe  and  pi  the  absolute 

pressure  per  sq.  foot  at  the  top  of  the  suction  pipe,  then  ~-~ 
is  the  total  centrifugal  head  hc. 


1000  1800  200Q  2200 

per  Minute. 

Fig.  282. 


2400 


A  curve  may  now  be  plotted  similar  to  that  shown  in  Fig.[282 
which  has  been  drawn  from  data  obtained  from  the  pump,  shown 
in  Fig.  275. 

When  the  head  is  44  feet,  the  speed  at  which  delivery  would 
just  start  is  2000  revolutions  per  minute. 

On  reference  to  Fig.  293,  which  shows  the  discharge  under 
different  heads  at  various  speeds,  the  discharge  at  2000  revolutions 
per  minute  when  the  head  is  44  feet  is  seen  to  be  12  cubic  feet 
per  minute.  This  means,  that  if  the  pump  is  to  discharge  against 
this  head  at  this  speed  it  cannot  deliver  less  than  12  cubic  feet 
per  minute. 

228.  Variation  of  the  discharge  of  a  centrifugal  pump 
with  the  head  when  the  speed  is  kept  constant*. 

Head-discharge  curve  at  constant  velocity.  If  the  speed  of  a 
centrifugal  pump  is  kept  constant  and  the  head  varied,  the  dis- 
charge varies  as  shown  in  Figs.  283,  285,  289,  and  292. 

*  See  also  page  418. 


CENTRIFUGAL    PUMPS 


411 


The  curve  No.  2,  of  Fig.  283,  shows  the  variation  of  the  head 
with  discharge  for  the  pump  shown  in  Fig.  275  when  running  at 
1950  revolutions  per  minute;  and  that  of  Fig.  285  was  plotted 
from  experimental  data  obtained  by  M.  Eateau  on  a  pump  having 
a  wheel  11*8  inches  diameter. 

The  data  for  plotting  the  curve  shown  in  Fig.  289*  was 
obtained  from  a  large  centrifugal  pump  having  a  spiral  chamber. 
In  the  case  of  the  dotted  curve  the  head  is  always  less  than  the 
centrifugal  head  when  the  flow  is  zero,  and  the  discharge  against 
a  given  head  has  only  one  value. 
10 


Fig.  283. 


1  2  3  4 

Radii  Velocity  of  flow  from,  Wheel. 
Head-discharge  curve  for  Centrifugal  Pump.     Velocity  Constant. 


Fig.  284.     Velocity-discharge  curve  for  Centrifugal  Pump.     Head  Constant. 

In  Fig.  285  the  discharge  when  the  head  is  80  feet  may  be 
either  '9  or  3'5  cubic  feet  per  minute.  The  work  required  to  drive 
the  pump  will  be  however  very  different  at  the  two  discharges, 
and,  as  shown  by  the  curves  of  efficiency,  the  actual  efficiencies 
for  the  two  discharges  are  very  different.  At  the  given  velocity 
therefore  and  at  80  feet  head,  the  flow  is  ambiguous  and  is 
unstable,  and  may  suddenly  change  from  one  value  to  the  other, 
or  it  may  actually  cease,  in  which  case  the  pump  would  not  start 
again  without  the  velocity  vl  being  increased  to  70'7  feet  per 
second.  This  value  is  calculated  from  the  equation 

"  =80', 


*  Proceedings  Inst.  Mech.  Engs.,  1903. 


412 


HYDRAULICS 


the  coefficient  m  for  this  pump  being  1'02.  For  the  flow  to  be 
stable  when  delivering  against  a  head  of  80  feet,  the  pump  should 
be  run  with  a  rim  velocity  greater  than  70' 7  feet  per  second,  in 
which  case  the  discharge  cannot  be  less  than  4J  cubic  feet  per 
minute,  as  shown  by  the  velocity-discharge  curve  of  Fig.  287. 
The  method  of  determining  this  curve  is  discussed  later. 

Pomp  Wheel 


90 
80 

^  

—  ^ 

^^^ 

^ 

^'l° 
$60 

$*° 
'*  4V 

^30 
^20 
tn\ 

^1-02  V?  jr 
?,n    1   Hl 

ad-  Disci 

large  Car 

ye 

^ 

v,  =  6G\ 

oersec. 

1                     J 

1               3               4 

Fig.  285. 


. 

Discharge  in,  c.ft.  per  mini. 


Fig.  286. 


75 


60 


/ 

\^ 

Velocity-] 

discharge 
Constant  - 

Curve  ^ 

/ 

^~"^~~~-.  . 

—  

1               Z               3               4 

Fig.  287. 


Example.  A  centrifugal  pump,  when  discharging  normally,  has  a  peripheral 
velocity  of  50  feet  per  second. 

The  angle  0  at  exit  is  30  degrees  and  the  manometric  efficiency  is  60  per  cent. 
The  radial  velocity  of  flow  at  exit  is  2^/T. 

Determine  the  lift  h  and  the  velocity  of  the  wheel  at  which  it  will  start  delivery 
under  full  head. 


V  =  50  -(2jh)  cos  130 
=  50-1-73^. 


CENTRIFUGAL   PUMPS  413 

,    (50  -  1-73  Jh)  50 
Therefore  ft  =  0-6.  -  —  , 

from  which  ft  =  37  feet. 

Let  v2  be  the  velocity  of  the  rim  of  the  wheel  at  which  pumping  commences. 
Then  assuming  the  centrifugal  head,  when  there  is  no  discharge,  is 
v  '* 

l=*> 

i?2=:48-6  ft.  per  sec. 

229.     Bernouilli's  equations  applied  to  centrifugal  pumps. 

Consider  the  motion  of  the  water  in  any  passage  between  two 
consecutive  vanes  of  a  wheel.  Let  p  be  the  pressure  head  at 
inlet,  pi  at  outlet  and  pa  the  atmospheric  pressure  per  sq.  foot. 

If  the  wheel  is  at  rest  and  the  water  passes  through  it  in 
the  same  way  as  it  does  when  the  wheel  is  in  motion,  and  all 
losses  are  neglected,  and  the  wheel  is  supposed  to  be  horizontal,  by 
Bernouilli's  equations  (see  Figs.  277  and  278), 

Pi  +  ^  -  P  +  In2  m 

w     2g     w     2g 

But  since,  due  to  the  rotation,  a  centrifugal  head 


is  impressed  on  the  water  between  inlet  and  outlet,  therefore, 


-  +- 

w     2g     w      2g      2g     2g  ' 

P!       p       V*       V*       Vr2      V? 

«-S=2T2^-2»  ..................  (4)' 

From  (3)  by  substitution  as  on  page  337, 

a+g,.*  +  g+2ifl±S          ......(5), 

w      20      w      2g        g         g 
and  when  U  is  radial  and  therefore  equal  to  u, 


_ 

w      2g      w     2g        g 

If  now  the  velocity  Ui  is  diminished  gradually  and  without 
shock,  so  that  the  water  leaves  the  delivery  pipe  with  a  velocity 
udy  and  if  frictional  losses  be  neglected,  the  height  to  which  the 
water  can  be  lifted  above  the  centre  of  the  pump  is,  by  Bernouilli's 
equation, 

»-5*g-57j{  .....................  0). 

If  the  centre  of  the  wheel  is  h0  feet  above  the  level  of  the  water 
in  the  sump  or  well,  and  the  water  in  the  well  is  at  rest, 

*=  =  *.  +  *+£  ...(8). 

w  w     2 


414  HYDRAULICS 

Substituting  from  (7)  and  (8)  in  (6) 


ll  7^7        ^        d 

-  —  fi  +  ho  +  TJ- 
9  22<7 

=  H0+g=H    .....................  (9). 

This  result  verifies  the  fundamental  equation  given  on  page  398. 
Further  from  equation  (6) 


Example.  The  centre  of  a  centrifugal  pump  is  15  feet  above  the  level  of  the 
water  in  the  sump.  The  total  lift  is  60  feet  and  the  velocity  of  discharge  from  the 
delivery  pipe  is  5  feet  per  second.  The  angle  <£  at  discharge  is  135  degrees,  and 
the  radial  velocity  of  flow  through  the  wheel  is  5  feet  per  second.  Assuming  there 
are  no  losses,  find  the  pressure  head  at  the  inlet  and  outlet  circumferences. 

At  inlet  £  =  34'-15'-! 

=  18-6  feet. 
The  radial  velocity  at  outlet  is 

wx  =  5  feet  per  second, 


and  ^ 

and  therefore,  ^2  +  5^  =  1940  .......................................  (1), 

from  which  »1  =  41-6  feet  per  second, 

and  Vj 


The  pressure  head  at  outlet  is  then 

ft^  +  tt'-HL1 

w      w  2g 

=  45  feet. 
To  find  the  velocity  vl  when  <j>  is  made  30  degrees. 


therefore  (1)  becomes  v-f  -  5  */3  .  vx  =  1940, 

from  which  ^  =  48-6  ft.  per  sec. 

and  V1  =  40 

Then  Hi.  =  25-4  feet,  and  ^  =  53-6  feet. 

2g  w 

230.     Losses  in  centrifugal  pumps. 

The  losses  of  head  in  a  centrifugal  pump  are  due  to  the  same 
causes  as  the  losses  in  a  turbine. 

Loss  of  head  at  exit.  The  velocity  Ui  with  which  the  water 
leaves  the  wheel  is,  however,  usually  much  larger  than  in  the 
case  of  the  turbine,  and  as  it  is  not  an  easy  matter  to  diminish 
this  velocity  gradually,  there  is  generally  a  much  larger  loss  of 
velocity  head  at  exit  from  the  wheel  in  the  pump  than  in  the 
turbine. 


CENTRIFUGAL   PUMPS  415 

In  many  of  the  earlier  pumps,  which  had  radial  vanes  at  exit, 

IT  2 
the  whole  of  the  velocity  head  ~  was  lost,  no  special  precautions 

being  taken  to  diminish  it  gradually  and  the  efficiency  was 
constantly  very  low,  being  less  than  40  per  cent. 

The  effect  of  the  angle  <f>  on  the  efficiency  of  the  pump.  To 
increase  the  efficiency  Appold  suggested  that  the  blade  should  be 
set  back,  the  angle  <f>  being  thus  less  than  90  degrees,  Fig.  272. 

Theoretically,  the  effect  on  the  efficiency  can  be  seen  by 
considering  the  three  cases  considered  in  section  220  and  illustrated 

U  2 

in  Fig.  279.     When  <f>  is  90  degrees  ~~  is  '54H,  and  when  <£  is 

U  2 
30  degrees  ~  is  '36H.     If,  therefore,  in  these  two  cases  this  head 

*j 

is  lost,  while  the  other  losses  remain  constant,  the  efficiency  in 
the  second  case  is  18  per  cent,  greater  than  in  the  first,  and  the 
efficiencies  cannot  be  greater  than  46  per  cent,  and  64  per  cent. 
respectively. 

In  general  when  there  is  no  precaution  taken  to  utilise  the 
energy  of  motion  at  the  outlet  of  the  wheel,  the  theoretical  lift  is 


and  the  maximum  possible  manometric  efficiency  is 


Substituting  for  Yi,  Vi-  Ui  cot  <£,  and  for  Ui2,  Yi2  +  ^i2, 


Vi        U? 

= 


and 


2  (vi  —  vlul  cot 
—  u^  cosec2  <f> 


vi  vi  —  Ui  cot  < 

When  vl  is  30  feet  per  second,  u^  5  feet  per  second  and  <£ 
150  degrees,  e  is  56  per  cent,  and  when  <f>  is  90  degrees  e  is 
48'5  per  cent. 

Experiments  also  show  that  in  ordinary  pumps  for  a  given  lift 
and  discharge  the  efficiency  is  greater  the  smaller  the  angle  <£. 

Parsons*  found  that  when  <£  was  90  degrees  the  efficiency  of  a 
pump  in  which  the  wheel  was  surrounded  by  a  circular  casing 
was  nearly  10  per  cent,  less  than  when  the  angle  <f>  was  made 
about  165  degrees. 

*  Proceedings  List.  C.  E.,  Vol.  XLVII.  p.  272. 


416  HYDRAULICS 

Stanton  found  that  a  pump  7  inches  diameter  having  radial 
vanes  at  discharge  had  an  efficiency  of  8  per  cent,  less  than  when 
the  angle  <f>  at  delivery  was  150  degrees.  In  the  first  case  the 
maximum  actual  efficiency  was  only  39'6  per  cent.,  and  in  the 
second  case  50  per  cent. 

It  has  been  suggested  by  Dr  Stanton  that  a  second  reason  for 
the  greater  efficiency  of  the  pump  having  vanes  curved  back  at 
outlet  is  to  be  found  in  the  fact  that  with  these  vanes  the  variation 
of  the  relative  velocity  of  the  water  and  the  wheel  is  less  than 
when  the  vanes  are  radial  at  outlet.  It  has  been  shown  experi- 
mentally that  when  the  section  of  a  stream  is  diverging,  that  is 
the  velocity  is  diminishing  and  the  pressure  increasing,  there  is 
a  tendency  for  the  stream  lines  to  flow  backwards  towards  the 
sections  of  least  pressure.  These  return  stream  lines  cause  a  loss 
of  energy  by  eddy  motions.  Now  in  a  pump,  when  the  vanes  are 
radial,  there  is  a  greater  difference  between  the  relative  velocity 
of  the  water  and  the  vane  at  inlet  and  outlet  than  when  the  angle 
<£  is  less  than  90  degrees  (see  Fig.  279),  and  it  is  probable  there- 
fore that  there  is  more  loss  by  eddy  motions  in  the  wheel  in  the 
former  case. 

Loss  of  head  at  entry.  To  avoid  loss  of  head  at  entry  the  vane 
must  be  parallel  to  the  relative  velocity  of  the  water  and  the 
vane. 

Unless  guide  blades  are  provided  the  exact  direction  in  which 
the  water  approaches  the  edge  of  the  vane  is  not  known.  If  there 
were  no  friction  between  the  water  and  the  eye  of  the  wheel  it 
would  be  expected  that  the  stream  lines,  which  in  the  suction  pipe 
are  parallel  to  the  sides  of  the  pipe,  would  be  simply  turned  to 
approach  the  vanes  radially. 

It  has  already  been  seen  that  when  there  is  no  flow  the  water 
in  the  eye  of  the  wheel  is  made  to  rotate  by  friction,  and  it  is 
probable  that  at  all  flows  the  water  has  some  rotation  in  the  eye 
of  the  wheel,  but  as  the  delivery  increases  the  velocity  of  rotation 
probably  diminishes.  If  the  water  has  rotation  in  the  same 
direction  as  the  wheel,  the  angle  of  the  vane  at  inlet  will  clearly 
have  to  be  larger  for  no  shock  than  if  the  flow  is  radial.  That 
the  water  has  rotation  before  it  strikes  the  vanes  seems  to  be 
indicated  by  the  experiments  of  Mr  Livens  on  a  pump,  the  vanes 
of  which  were  nearly  radial  at  the  inlet  edge.  (See  section  236.) 
The  efficiencies  claimed  for  this  pump  are  so  high,  that  there 
could  have  been  very  little  loss  at  inlet. 

If  the  pump  has  to  work  under  variable  conditions  and  the 
water  be  assumed  to  enter  the  wheel  at  all  discharges  in  the  same 
direction,  the  relative  velocity  of  the  water  and  the  edge  of  the 


CENTRIFUGAL   PUMPS  417 

vane  can  only  be  parallel  to  the  tip  of  the  vane  for  one  discharge, 
and  at  other  discharges  in  order  to  make  the  water  move  along 
the  vane  a  sudden  velocity  must  be  impressed  upon  it,  which 
causes  a  loss  of  energy. 

Let  2^2,  Fig.  288,  be  the  velocity  with  which  the  water  enters  a 
wheel,  and  0  and  v  the  inclination 
and  velocity  of  the  tip  of  the  vane         \e--u*  ->j 
at  inlet  respectively. 

The  relative  velocity  of  u%  and  v 
is  V/,  the  vector  difference  of  u% 
and  v. 

The  radial  component  of  flow 
through  the  opening  of  the  wheel 
must  be  equal  to  the  radial  com- 
ponent of  u2,  and  therefore  the 
relative  velocity  of  the  water  along  the  tip  of  the  vane  is  Vr. 

If  u*  is  assumed  to  be  radial,  a  sudden  velocity 

u8  -  v  —  u?,  cot  0 
has  thus  to  be  given  to  the  water. 

If  Uz  has  a  component  in  the  direction  of  rotation  us  will  be 
diminished. 

It  has  been  shown  (page  67),  on  certain  assumptions,  that  if 
a  body  of  water  changes  its  velocity  from  va  to  vd  suddenly,  the 

head  lost  is  ^-^  —     ,  or  is  the  head  due  to  the  change  of  velocity. 
In  this  case  the  change  of  velocity  is  u8,  and  the  head  lost  may 

JfU2 

reasonably  be  taken  as  -n~~  •  If  &  is  assumed  to  be  unity,  the 
effective  work  done  on  the  water  by  the  wheel  is  diminished  by 


If  now  this  loss  takes  place  in  addition  to  the  velocity  head 
being  lost  outside  the  wheel,  and  friction  losses  are  neglected, 
then 


g        20  20 


Zg 


V* 


L.  H.  27 


418 


HYDRAULICS 


Example.  The  radial  velocity  of  flow  through  a  pump  is  5  feet  per  second. 
The  angle  0  is  30  degrees  and  the  angle  d  is  15  degrees.  The  velocity  of  the 
outer  circumference  is  50  feet  per  sec.  and  the  radius  is  twice  that  of  the  inner 
circumference. 

Find  the  theoretical  lift  on  the  assumption  that  the  whole  of  the  kinetic  energy 
is  lost  at  exit. 


0 

2g 

=  37-5  feet. 

The  theoretical  lift  neglecting  all  losses  is  64-2  feet,  and  the  manometric 
efficiency  is  therefore  58  per  cent. 

231.  Variation  of  the  head  with  discharge  and  with  the 
speed  of  a  centrifugal  pump. 

It  is  of  interest  to  study  by  means  of  equation  (1),  section  230, 
the  variation  of  the  discharge  Q  with  the  velocity  of  the  pump 
when  h  is  constant,  and  the  variation  of  the  head  with  the 
discharge  when  the  velocity  of  the  pump  is  constant,  and  to 
compare  the  results  with  the  actual  results  obtained  from 
experiment. 

The  full  curve  of  Fig.  289  shows  the  variations  of  the  head 
with  the  discharge  when  the  velocity  of  a  wheel  is  kept  constant. 
The  data  for  which  the  curve  has  been  plotted  is  indicated  in 
the  figure. 


JO 

n 

13 

I. 
1" 

$10 

>< 

**">. 

*""•*•» 

^ 

\ 

/ 

\ 

\ 

/ 

Z% 

(f=9t 

e-15 

orma 

i 

)Ft.p< 
)° 

0 

zrSet 

>. 

\ 

\ 

/ 

\ 

N 
o 

1  rcudbiaJL  velocity  =1i 

2       I*        l»      Is'*" 

rir 

velocity  of  FLow=  Q 

A, 

Fig.  289.     Head-discharge  curve  at  constant  velocity. 
When  the  discharge  is  zero 


The  velocity  of  flow  -£-  at  outlet  has  been  assumed  equal  to 

AI 

9-  at  inlet. 

Values  of  1,  2,  etc.  were  given  to  ~  and  the  corresponding 

.XX 

values  of  h  found  from  equation  (1). 


CENTRIFUGAL  PUMPS  419 

When  the  discharge  is  normal,  that  is,  the  water  enters  the 
wheel  without  shock,  -^  is  4  feet  and  h  is  14  feet.  The  theoretical 
head  assuming  no  losses  is  then  28  feet  and  the  manometric 
efficiency  is  thus  50  per  cent.  For  less  or  greater  values  of  -£ 

the  head  diminishes  and  also  the  efficiency. 

The  curve  of  Fig.  290  shows  how  the  flow  varies  with  the 
velocity  for  a  constant  value  of  h,  which  is  taken  as  12  feet. 


2  fee 


RadiaL  Velocity  througk  Wheel. 

Fig.  290.     Velocity-discharge  curve  at  constant  head  for  Centrifugal  Pump. 

It  will  be  seen  that  when  the  velocity  Vi  is  31'9  feet  per  second 
the  velocity  of  discharge  may  be  either  zero  or  8*2  feet  per  second. 
This  means  that  if  the  head  is  12  feet,  the  pump,  theoretically, 
will  only  start  when  the  velocity  is  31 '9  feet  per  second  and  the 
velocity  of  discharge  will  suddenly  become  8*2  feet  per  second. 
If  now  the  velocity  Vi  is  diminished  the  pump  still  continues  to 
discharge,  and  will  do  so  as  long  as  vl  is  greater  than  26'4  feet  per 
second.  The  flow  is  however  unstable,  as  at  any  velocity  vc  it  may 
suddenly  change  from  CE  to  CD,  or  it  may  suddenly  cease,  and  it 
will  not  start  again  until  Vi  is  increased  to  31*9  feet  per  second. 

232.  The  effect  of  the  variation  of  the  centrifugal  head 
and  the  loss  by  friction  on  the  discharge  of  a  pump. 

If  then  the  losses  at  inlet  and  outlet  were  as  above  and  were 
the  only  losses,  and  the  centrifugal  head  in  an  actual  pump  was 
equal  to  the  theoretical  centrifugal  head,  the  pump  could  not  be 
made  to  deliver  water  against  the  normal  head  at  a  small  velocity 
of  discharge.  In  the  case  of  the  pump  considered  in  section  231, 
it  could  not  safely  be  run  with  a  rim  velocity  less  than  31*9  ft. 
per  sec.,  and  at  any  greater  velocity  the  radial  velocity  of  flow 
could  not  be  less  than  8  feet  per  second. 

27—2 


420  HYDRAULICS 

In  actual  pumps,  however,  it  has  been  seen  that  the  centrifugal 
head  at  commencement  is  greater  than 


There  is  also  loss  of  head,  which  at  high  velocities  and  in  small 
pumps  is  considerable,  due  to  friction.  These  two  causes  consider- 
ably modify  the  head-discharge  curve  at  constant  velocity  and  the 
velocity-discharge  curve  at  constant  head,  and  the  centrifugal 
head  at  the  normal  speed  of  the  pump  when  the  discharge  is  zero, 
is  generally  greater  than  any  head  under  which  the  pump  works, 
and  many  actual  pumps  can  deliver  variable  quantities  of  water 
against  the  head  for  which  they  are  designed. 

The  centrifugal  head  when  the  flow  is  zero  is 


2?  ' 

m  being  generally  equal  to,  or  greater  than  unity.  As  the  flow 
increases,  the  velocity  of  whirl  in  the  eye  of  the  wheel  and  in 
the  casing  will  diminish  and  the  centrifugal  head  will  therefore 
diminish. 

Let  it  be  assumed  that  when  the  velocity  of  flow  is  u  (supposed 
constant)  the  centrifugal  head  is 

(lev  —  nu)2 


Jc  and  n  being  constants  which  must  be  determined  by  experiment. 
When  u  is  zero 

V  W 


2g     2g      2g  ~  2g  > 

and  if  m  is  known  k  can  at  once  be  found. 

Let  it  further  be  assumed  that  the  loss  by  friction*  and  eddy 

motions,  apart  from  the  loss  at  inlet  and  outlet  is  ^  . 

*  The  loss  of  head  by  friction  will  no  doubt  depend  not  only  upon  u  but  also 
upon  the  velocity  ^  of  the  wheel,  and  should  be  written  as 

Cw2     C 


.... 

If  it  be  supposed  it  can  be  expressed  by  the  latter,  then  the  correction 
Wv*  _  2nkulVl     k^ 
2<7  20       ~~27' 

if  proper  values  are  given  to  fc,  nx  and  klt  takes  into  account  the  variation  of  the 
centritugal  head  and  also  the  friction  head  v,  . 


CENTRIFUGAL   PUMPS  421 


The  gross  head  h  is  then, 

,      v?     v2      u*  ,     2vu  cot  0 


cV 


If  now  the  head  h  and  flow  Q  be  determined  experimentally, 
the  difference  between  h  as  determined  from  equation  (1),  page  4J  7, 
and  the  experimental  value  of  h,  must  be  equal  to 

it2  ,  2       2, 

~(n~ 


ki  being  equal  to  (c2  -  n2). 

The  coefficient  k  being  known  from  an  experiment  when  u  is 
zero,  two  other  experiments  giving  corresponding  values  of  h  and 
u  will  determine  the  coefficients  n  and  ki  . 

The  head-discharge  curve  at  constant  velocity,  for  a  pump  such 
as  the  one  already  considered,  would  approximate  to  the  dotted 
curve  of  Fig.  289.  This  curve  has  been  plotted  from  equation  (2), 
by  taking  k  as  0*5,  n  as  7'64  and  ki  as  -  38. 

Substituting  values  for  k,  n,  kl}  cosec  <£  and  cot  <£,  equation  (2) 
becomes 

,      mv?     Cu  .  v 

h=      ~~ 


C  and  Ci  being  new  coefficients  ;  or  it  may  be  written 

h  =  ™l  +  ^  +  C3Q>  .....................  (4), 

Q  being  the  flow  in  any  desired  units,  the  coefficients  C2  and  C3 
varying  with  the  units.  If  equation  (4)  is  of  the  correct  form, 
three  experiments  will  determine  the  constants  m,  C2  and  C3 
directly,  and  having  given  values  to  any  two  of  the  three 
variables  h,  v,  and  Q  the  third  can  be  found. 

233.  The  effect  of  the  diminution  of  the  centrifugal  head 
and  the  increase  of  the  friction  head  as  the  flow  increases,  on 
the  velocity-discharge  curve  at  constant  head. 

Using  the  corrected  equation  (2),  section  232,  and  the  given 
values  of  k,  nt  and  ki  the  dotted  curve  of  Fig.  290  has  been  plotted. 

From  the  dotted  curve  of  Fig.  289  it  is  seen  that  u  cannot 
be  greater  than  5  feet  when  the  head  is  12  feet,  and  therefore  the 
new  curve  of  Fig.  290  is  only  drawn  to  the  point  where  u  is  5. 

The  pump  starts  delivering  when  v  is  27*7  feet  per  second  and 
the  discharge  increases  gradually  as  the  velocity  increases. 


422  HYDRAULICS 

The  pump  will  deliver,  therefore,  water  under  a  head  of 
12  feet  at  any  velocity  of  flow  from  zero  to  5  feet  per  second. 

In  such  a  pump  the  manometric  efficiency  must  have  its 
maximum  value  when  the  discharge  is  zero  and  it  cannot  be 

greater  than 

my* 


Via  -  ViUi  CQt  0  ' 


9 

This  is  the  case  with  many  existing  pumps  and  it  explains  why, 
when  running  at  constant  speed,  they  can  be  made  to  give  any 
discharge  varying  from  zero  to  a  maximum,  as  the  head  is 
diminished. 

234.     Special  arrangements  for  converting  the  velocity 

U2 
head  =-  with  which  the  water  leaves  the  wheel  into  pressure 

head. 

The  methods  for  converting  the  velocity  head  with  which  the 
water  leaves  the  wheel  into  pressure  head  have  been  indicated  on 
page  394.  They  are  now  discussed  in  greater  detail. 

Thomson's  vortex  or  whirlpool  chamber.  Professor  James 
Thomson  first  suggested  that  the  wheel  should  be  surrounded  by 
a  chamber  in  which  the  velocity  of  the  water  should  gradually 
change  from  Ui  to  ud  the  velocity  of  flow  in  the  pipe.  Such  a 
chamber  is  shown  in  Fig.  274.  In  this  chamber  the  water  forms 
a  free  vortex,  so  called  because  no  impulse  is  given  to  the  water 
while  moving  in  the  chamber. 

Any  fluid  particle  ab,  Fig.  281,  may  be  considered  as  moving 
in  a  circle  of  radius  r0  with  a  velocity  v0  and  to  have  also  a 
radial  velocity  u  outwards. 

Let  it  be  supposed  the  chamber  is  horizontal. 

If  W  is  the  weight  of  the  element  in  pounds,  its  momentum 

perpendicular   to   the    radius  is   —  -   and  the   moment   of    mo- 

9 

mentum  or  angular  momentum  about  the  centre  C  is  —  —  . 

For  the  momentum  of  a  body  to  change,  a  force  must  act  upon 
it,  and  for  the  moment  of  momentum  to  change,  a  couple  must  act 
upon  the  body. 

But  since  no  turning  effort,  or  couple,  acts  upon  the  element 
after  leaving  the  wheel  its  moment  of  momentum  must  be 
constant. 


CENTRIFUGAL   PUMPS  423 

Therefore, 

is  constant  or  v0r0  =  constant. 

If  the  sides  of  the  chamber  are  parallel  the  peripheral  area  of 
the  concentric  rings  is  proportional  to  r0,  and  the  radial  velocity  of 
flow  u  for  any  ring  will  be  inversely  proportional  to  r0,  and  there- 
fore, the  ratio  --  is  constant,  or  the  direction  of  motion  of  any 

element  with  its  radius  r0  is  constant,  and  the  stream  lines  are 
equiangular  spirals. 

If  no  energy  is  lost,  by  friction  and  eddies,  Bernoulli's  theorem 
will  hold,  and,  therefore,  when  the  chamber  is  horizontal 

U2  VQ  PQ 

2g  +  2g+w 

is  constant  for  the  stream  lines. 

This  is  a  general  property  of  the  free  vortex. 
If  u  is  constant 

V«        V0 

~-  +  —  =  constant. 
2g     w 

Let  the  outer  radius  of  the  whirlpool  chamber  be  R«,  and 
the  inner  radius  rll}.  Let  vr,w  and  vRw  be  the  whirling  velocities 
at  the  inner  and  outer  radii  respectively. 

Then  since  v0r0  is  a  constant, 

and  — °  +  ^-  =  constant, 

w      £g 

w  ~  w       2g       2g 


w 
When  R*  =  2rv 


w       w      4*  2g 

If  the  velocity  head  which  the  water  possesses  when  it  leaves 
the  vortex  chamber  is  supposed  to  be  lost,  and  hi  is  the  head  of 
water  above  the  pump  and  pa  the  atmospheric  pressure,  then 
neglecting  friction 

P*v_  ,  U<?       Pa 

—    fl/l   ~r     ~,    ~    T  . 

w  2g     w 


or 


w       2g      w' 


424  HYDRAULICS 

If  then  hQ  is  the  height  of  the  pump  above  the  well,  the  total 
lift  h*  is  hi  +  ho. 
Therefore, 


also  Prw  =  PI?  fw  —  R,  and  vrw  = 

Therefore 


But  from  equation  (6)  page  413, 


w     w     2g        g        2g  ' 
Therefore 

R2 


This  might  have  been  written  down  at  once  from  equation  (1), 

section  230.     For   clearly  if  there  is   a  gain   of   pressure  head 

•y  2    e          j^2  \ 

in  the  vortex  chamber  of  -^  (  1  —  -5-=  )  ,   the    velocity  head  to 

£g  \        n,w  / 

be  lost  will  be  less  by  this  amount  than  when  there  is  no  vortex 
chamber. 

Substituting  for  Vi  and  Ui  the  theoretical  lift  h  is  now 


,  _   i  -   ii  cot  <^>     u^     fa  -  HI  cot  <^>)2  E2  x1x 

—        ~%-        ~W     "'B? 
When  the  discharge  or  rim  velocity  is  not  normal,  there  is  a 
further  loss  of  head  at  entrance  equal  to 


«  - 


and 

h  = 


(2). 


When  there  is  no  discharge  vrw  is  equal  to  t?i  and 

V 


CENTRIFUGAL   PUMPS  425 


If  R  =  JR«,  and  v  =  \v\ , 

O  2 

Correcting  equation  (1)  in  order  to  allow  for  the  variation  of 
the  centrifugal  head  with  the  discharge,  and  the  friction  losses, 

7   _  Vi2  —  V\Ui  COt  <£       U\         (v\  —  Ksi  COt  <£)2  R2 

~g~    ~%9~ 


_  (v  —  u  cot  O)2     fcV  _  2nkuvi  _  kiuz 

~W~     "~W~  ~W  ""20"' 

2          /~1   /^\  /*^   /^\2 

1   •    r  i  7         WUi          (jz^JV        U.sU 

which  reduces  to         h  —  —c  —  +    ~      +    ^      . 

2g         2g         2g 

The  experimental  data  on  the  value  of  the  vortex  chamber 
per  86t  in  increasing  the  efficiency  is  very  limited. 

Stanton*  showed  that  for  a  pump  having  a  rotor  7  inches 
diameter  surrounded  by  a  parallel  sided  vortex  chamber  18  inches 
diameter,  the  efficiency  of  the  chamber  in  converting  velocity  head 
to  pressure  head  was  about  40  per  cent.  It  is  however  questionable 
whether  the  design  of  the  pump  was  such  as  to  give  the  best  results 
possible. 

So  far  as  the  author  is  aware,  centrifugal  pumps  with  vortex 
chambers  are  not  now  being  manufactured,  but  it  seems  very 
probable  that  by  the  addition  of  a  well-designed  chamber  small 
centrifugal  pumps  might  have  their  efficiencies  considerably  in- 
creased. 

235.     Turbine  pumps. 

Another  method,  first  suggested  by  Professor  Reynolds,  and 
now  largely  used,  for  diminishing  the  velocity  of  discharge  Ui 
gradually,  is  to  discharge  the  water  from  the  wheel  into  guide 
passages  the  sectional  area  of  which  should  gradually  increase 
from  the  wheel  outwards,  Figs.  275  and  276,  and  the  tangents  to  the 
tips  of  the  guide  blades  should  be  made  parallel  to  the  direction 
of  Ux. 

The  number  of  guide  passages  in  small  pumps  is  generally  four 
or  five. 

If  the  guide  blades  are  fixed  as  in  Fig.  275,  the  direction  of 
the  tips  can  only  be  correct  for  one  discharge  of  the  pump, 
but  except  for  large  pumps,  the  very  large  increase  in  initial  cost 
of  the  pump,  if  adjustable  guide  blades  were  used,  as  well  as 
the  mechanical  difficulties,  would  militate  against  their  adoption. 

Single  wheel  pumps  of  this  type  can  be  used  up  to  a  head  of 
100  feet  with  excellent  results,  efficiencies  as  high  as  85  per  cent. 
*  Proceedings  Imt.  C.  E.,  1903. 


426  HYDRAULICS 

having  been  claimed.  They  are  now  being  used  to  deliver  water 
against  heads  of  over  350  feet,  and  M.  Rateau  has  used  a  single 
wheel  3*16  inches  diameter  running  at  18,000  revolutions  per 
minute  to  deliver  against  a  head  of  936  feet. 

Loss  of  head  at  the  entrance  to  the  guide  passages.  If  the 
guide  blades  are  fixed,  the  direction  of  the  tips  can  only  be  correct 
for  one  discharge  of  the  pump.  For  any  other  discharge  than  the 
normal,  the  direction  of  the  water  as  it  leaves  the  wheel  is  not 
parallel  to  the  fixed  guide  and  there  is  a  loss  of  head  due  to 
shock. 

Let  cc  be  the  inclination  of  the  guide  blade  and  </>  the  vane 
angle  at  exit. 

Let  1*1  be  the  radial  velocity  of 
flow.  Then  BE,  Fig.  291,  is  the 
velocity  with  which  the  water  leaves 
the  wheel. 

The  radial  velocity  with  which 
the  water  enters  the  guide  passages  must  be  Ui  and  the  velocity 
along  the  guide  is,  therefore,  BF. 

There  is  a  sudden  change  of  velocity  from  BE  to  BF,  and  on 
the  assumption  that  the  loss  of  head  is  equal  to  the  head  due  to  the 
relative  velocity  FE,  the  head  lost  is 

fa  -  Ui  cot  <j>  —  ui  cot  «)2 

~W~ 
At  inlet  the  loss  of  head  is 

(v  -  u  cot  0)2 


and  the  theoretical  lift  is 

-,  _  Vi  -  ViUi  cot  <fr  _  (v-u  cot  6>)2  _  (^  —  Ui  cot  <£  -  Ui  cot  a) 
~~  ~ 


__  l  cot  a     2vu  cot  6 

2g     2g  +        2g  2g 

u?  (cot  <£  +  cot  a)2     u2  cot2  0 

2g  2g      ......  (1)' 

To  correct  for  the  diminution  of  the  centrifugal  head  and  to 
allow  for  friction, 

feV     2^71  .  ^          u^ 
20  ~        2<f~          I2g> 
must  be  added,  and  the  lift  is  then 

h  =  ^  -  ^-  +  ??MM5ota     2vu  cot  6     ui*  (cot  <j>  +  cot  a)2 
20     20  20  2g  ~^g~ 

2knvlUl 


_  _ 

2g          2g          2g       ~  2g 


CENTRIFUGAL   PUMPS 


427 


which,  since  u  can  always  be  written  as  a  multiple  of  t&i,  reduces 
to  the  form 

2gh  =  mvl*  +  Culv1  +  Clu1*  .....................  (2). 

Equations  for  the  turbine  pump  shown  in  Fig.  275.    Character- 
istic curves.     Taking  the  data 

0=   5  degrees,  cot<9  =  ll'43 
</>  =  30       „        cot<£  =   1*732 


a=   3 


cota=19'6 


equation  (2)  above  becomes 

2gh  =  ' 


eo- 


(3). 


irv  Cubic  Feet  per  Itfinjuute. 


01  34-5 

JLculiaJj  Velocity  cub  Eacib  from,  fheWheeU.  Teet  per Sec onaU. 

Fig.  292.     Head-discharge  curves  at  constant  speed  for  Turbine  Pump. 

From  equation  (3)  taking  Vi  as  50  feet  per  second,  the  head- 
discharge  curve  No.  1,  of  Fig.  283,  has  been  drawn,  and  taking  h 
as  35  feet,  the  velocity-discharge  curve  No.  1,  of  Fig.  284,  has  been 
plotted. 

In  Figs.  292 — 4  are  shown  a  sanies  of  head-discharge  curves  at 


428 


HYDRAULICS 


constant  speed,  velocity-discharge  curves  at  constant  head,  and 
head-velocity  curves  at  constant  discharge,  respectively. 

The  points  shown  near  to  the  curves  were  determined  experi- 
mentally, and  the  curves,  it  will  be  seen,  are  practically  the  mean 
curves  drawn  through  the  experimental  points.  They  were  how- 
ever plotted  in  all  cases  from  the  equation 

2gh  =  1-087^  +  2'26tM?i  -  62' W, 

obtained  by  substituting  for  m,  C  and  d  in  equation  (2)  the  values 
1*087,  2'26  and  -  62'1  respectively.  The  value  of  m  was  obtained 
by  determining  the  head  h,  when  the  stop  valve  was  closed,  for 
speeds  between  1500  and  2500  revolutions  per  minute,  Fig.  282. 
The  values  of  C  and  Ci  were  first  obtained,  approximately,  by 
taking  two  values  of  u^  and  vt  respectively  from  one  of  the 
actual  velocity-discharge  curves  near  the  middle  of  the  series,  for 
which  h  was  known,  and  from  the  two  quadratic  equations  thus 
obtained  C  and  Ci  were  calculated.  By  trial  C  and  Ci  were  then 
corrected  to  make  the  equation  more  nearly  fit  the  remaining 
curves. 


mo  1900 . 

Speed—  ReYoluALon&  per  Mirvuube,. 

Fig.  293.     Velocity-Discharge  curves  at  Constant  Head. 

No  attempt  has  been  made  to  draw  the  actual  mean  curves  in 
the  figures,  as  in  most  cases  the  difference  between  them  and  the 
calculated  curves  drawn,  could  hardly  be  distinguished.  The 
reader  can  observe  for  himself  what  discrepancies  there  are  between 
the  mean  curves  through  the  points  and  the  calculated  curves.  It 


CENTRIFUGAL   PUMPS 


429 


will  be  seen  that  for  a  very  wide  range  of  speed,  head,  and 
discharge,  the  agreement  between  the  curves  and  the  observed 
points  is  very  close,  and  the  equation  can  therefore  be  used  with 
confidence  for  this  particular  pump  to  determine  its  performance 
under  stated  conditions. 

It  is  interesting  to  note,  that  the  experiments  clearly  indicated 
the  unstable  condition  of  the  discharge  when  the  head  was  kept 
constant  and  the  velocity  was  diminished  below  that  at  which  the 
discharge  commenced. 


20 


-2300- 


ieoo 


1900 


2000 

S    per 


2409 


Fig.  294.     Head-velocity  curves  at  Constant  Discharge. 

236.    Losses  in  the  spiral  casings  of  centrifugal  pumps. 

The  spiral  case  allows  the  mean  velocity  of  flow  toward  the 
discharge  pipe  to  be  fairly  constant  and  the  results  of  experiment 
seem  to  show  that  a  large  percentage  of  the  velocity  of  the  water 
at  the  outlet  of  the  wheel  is  converted  into  pressure  head. 
Mr  Livens*  obtained,  for  a  pump  having  a  wheel  19 \  inches 
diameter  running  at  550  revolutions  per  minute,  an  efficiency  of 
71  per  cent,  when  delivering  1600  gallons  per  minute  against  a 
head  of  25  feet.  The  angle  <£  was  about  13  degrees  and  the  mean 
of  the  angle  0  for  the  two  sides  of  the  vane  81  degrees. 

For  a  similar  pump  21f  inches  diameter  an  efficiency  of  82  per 
cent,  was  claimed. 


Proceedings  Inst.  Mech.  Engs.,  1903. 


430  HYDRAULICS 


The  author  finds  the  equation  to  the  head-discharge  curve  for 
the  19  J  inches  diameter  pump  from  Mr  Livens'  data  to  be 


and  for  the  21|  inches  diameter  pump 

Tl&Y5-  4-5^  =  2^   .....................  (2). 

The  velocity  of  rotation  of  the  water  round  the  wheel  will  be 
less  than  the  velocity  with  which  the  water  leaves  the  wheel  and 
there  will  be  a  loss  of  head  due  to  the  sudden  change  in  velocity. 

k  U  2 
Let  this  loss  of  head  be  written  -^-L  .     The  head,  when  ^  is  the 

radial  velocity  of  flow  at  exit  and  assuming  the  water  enters  the 
wheel  radially,  is  then 

_Viz-  ViUi  cot  <f>  _  fc3Ui2  _  (v-ucotO)2 

9  ~W  %9 

Taking  friction  and  the  diminution  of  centrifugal  head  into 
account, 

,  _v*—ViUiCot<l>     &3Ui2     (v  —  ucotOy 

~~i~      "W      ~W~ 

which  again  may  be  written 
,  _ 
= 


The  values  of  m}  C  and  Cj.  are  given  for  two  pumps  in  equations 
(1)  and  (2). 

237.     General  equation  for  a  centrifugal  pump. 

The  equations  for  the  gross  head  h  at  discharge  Q  as  determined 
for  the  several  classes  of  pumps  have  been  shown  to  be  of  the  form 

C3Q2 


or,  if  u  is  the  velocity  of  flow  from  the  wheel, 
,  _  mv*     Cuv     du* 
~-W"^9~~  2?  ' 

in  which  ra  varies  between  1  and  1*5.     The  coefficients  C2  and  C3 
for  any  pump  will  depend  upon  the  unit  of  discharge. 

As  a  further  example  and  illustrating  the  case  in  which  at 
certain  speeds  the   flow  may  be   unstable,   the   curves  of   Figs. 
285  —  287  may  be  now  considered.     When  Vi  is  66  feet  per  second 
the  equation  to  the  head  discharge  curve  is 
,  =  1-02^     15'5Q^ 

20  2g 

Q  being  in  cubic  feet  per  minute. 


CENTRIFUGAL   PUMPS  431 

The  velocity-discharge  curve  for  a  constant  head  of  80  feet  as 
calculated  from  this  equation  is  shown  in  Fig.  287. 

To  start  the  pump  against  a  head  of  80  feet  the  peripheral 
velocity  has  to  be  70*7  feet  per  second,  at  which  velocity  the 
discharge  Q  suddenly  rises  to  4*3  cubic  feet  per  minute. 

The  curves  of  actual  and  manometric  efficiency  are  shown  in 
Fig.  286,  the  maximum  for  the  two  cases  occurring  at  different 
discharges. 

238.  The  limiting  height  to  which  a  single  wheel  centri- 
fugal pump  can  be  used  to  raise  water. 

The  maximum  height  to  which  a  centrifugal  pump  can  raise 
water,  depends  theoretically  upon  the  maximum  velocity  at  which 
the  rim  of  the  wheel  can  be  run. 

It  has  already  been  stated  that  rim  velocities  up  to  250  feet 
per  second  have  been  used.  Assuming  radial  vanes  and  a  mano- 
metric efficiency  of  50  per  cent.,  a  pump  running  at  this  velocity 
would  lift  against  a  head  of  980  feet. 

At  these  very  high  velocities,  however,  the  wheel  must  be  of 
some  material  such  as  bronze  or  cast  steel,  having  considerable 
resistance  to  tensile  stresses,  and  special  precautions  must  be 
taken  to  balance  the  wheel.  The  hydraulic  losses  are  also 
considerable,  and  manometric  efficiencies  greater  than  50  per 
cent,  are  hardly  to  be  expected. 

According  to  M.  Eateau  *,  the  limiting  head  against  which  it  is 
advisable  to  raise  water  by  means  of  a  single  wheel  is  about 
100  feet,  and  the  maximum  desirable  velocity  of  the  rim  of  the 
wheel  is  about  100  feet  per  second. 

Single  wheel  pumps  to  lift  up  to  350  feet  are  however  being 
used.  At  this  velocity  the  stress  in  a  hoop  due  to  centrifugal  forces 
is  about  7250  Ibs.  per  sq.  incht. 

239.  The  suction  of  a  centrifugal  pump. 

The  greatest  height  through  which  a  centrifugal  or  other  class 
of  pump  will  draw  water  is  about  27  feet.  Special  precaution  has 
to  be  taken  to  ensure  that  all  joints  on  the  suction  pipe  are  perfectly 
air-tight,  and  especially  is  this  so  when  the  suction  head  is  greater 
than  15  feet;  only  under  special  circumstances  is  it  therefore  de- 
sirable for  the  suction  head  to  be  greater  than  this  amount,  and  it 
is  always  advisable  to  keep  the  suction  head  as  small  as  possible. 

*  "Pompes  Centrifuges,"  etc..  Bulletin  de  la  Societe  de  VIndustrie  minerale, 
1902 ;  Engineer,  p.  236,  March,  1902. 

f  See  Ewing's  Strength  of  Materials ;  Wood's  Strength  of  Structural  Members ; 
The  Steam  Turbine,  Stodola. 


432 


HYDRAULICS 


CENTRIFUGAL    PUMPS 


433 


240.     Series  or  multi-stage  turbine  pumps. 

It  has  been  stated  that  the  limiting  economical  head  for  a  single 
wheel  pump  is  about  100  feet,  and  for  high  heads  series  pumps 
are  now  generally  used. 


Fig.  296.     General  Arrangement  of  Worthington  Multi-stage  Turbine  Pump. 

By  putting  several  wheels  or  rotors  in  series  on  one  shaft,  each 
rotor  giving  a  head  varying  from  100  to  200  feet,  water  can  be 
lifted   to    practically  any  height,   and    such   pumps  have  been 
L.  H.  28 


434 


HYDRAULICS 


constructed  to  work  against  a  head  of  2000  feet.  The  number 
of  rotors,  on  one  shaft,  may  be  from  one  to  twelve  according 
to  the  total  head.  For  a  given  head,  the  greater  the  number  of 
rotors  used,  the  less  the  peripheral  velocity,  and  within  certain 
limits  the  greater  the  efficiency. 

Figs.  295  and  296  show  a  longitudinal  section  and  general 
arrangement,  respectively,  of  a  series,  or  multi-stage  pump,  as 
constructed  by  the  Worthington  Pump  Company.  On  the  motor 
shaft  are  fixed  three  phosphor-bronze  rotors,  alternating  with  fixed 
guides,  which  are  rigidly  connected  to  the  outer  casing,  and  to 
the  bearings.  The  water  is  drawn  in  through  the  pipe  at  the  left 
of  the  pump  and  enters  the  first  wheel  axially.  The  water  leaves 
the  first  wheel  at  the  outer  circumference  and  passes  along  an 
expanding  passage  in  which  the  velocity  is  gradually  diminished 
and  enters  the  second  wheel  axially.  The  vanes  in  the  passage 
are  of  hard  phosphor-bronze  made  very  smooth  to  reduce  friction 
losses  to  a  minimum.  The  water  passes  through  the  remaining 
rotors  and  guides  in  a  similar  manner  and  is  finally  discharged 
into  the  casing  and  thence  into  the  delivery  pipe. 


Fig.  297.     Sulzer  Multi-stage  Turbine  Pump. 

The  difference  in  pressure  head  at  the  entrances  to  any  two 
consecutive  wheels  is  the  head  impressed  on  the  water  by  one 
wheel.  If  the  head  is  h  feet,  and  there  are  n  wheels  the  total 
lift  is  nearly  nh  feet.  The  vanes  of  each  wheel  and  the  directions 
of  the  guide  vanes  are  determined  as  explained  for  the  single 
wheel  so  that  losses  by  shock  are  reduced  to  a  minimum,  and 
the  wheels  and  guide  passages  are  made  smooth  so  as  to  reduce 
friction. 

Through  the  back  of  each  wheel,  just  above  the  boss,  are 
a  number  of  holes  which  allow  water  to  get  behind  part  of  the 
wheel,  under  the  pressure  at  which  it  enters  the  wheel,  to  balance 
the  end  thrust  which  would  otherwise  be  set  up. 


CENTRIFUGAL   PUMPS  435 

The  pumps  can  be  arranged  to  work  either  vertically  or 
horizontally,  and  to  be  driven  by  belt,  or  directly  by  any  form 
of  motor. 

Fig.  297  shows  a  multi-stage  pump  as  made  by  Messrs  Sulzer. 
The  rotors  are  arranged  so  that  the  water  enters  alternately 
from  the  left  and  right  and  the  end  thrust  is  thus  balanced. 
Efficiencies  as  high  as  84  per  cent,  have  been  claimed  for  multi- 
stage pumps  lifting  against  heads  of  1200  feet  and  upwards. 

The  Worthington  Pump  Company  state  that  the  efficiency 
diminishes  as  the  ratio  of  the  head  to  the  quantity  increases,  the 
best  results  being  obtained  when  the  number  of  gallons  raised 
per  minute  is  about  equal  to  the  total  head. 

Example.  A  pump  is  to  be  driven  by  a  motor  at  1450  revolutions  per  minute,  and 
is  required  to  lift  45  cubic  feet  of  water  per  minute  against  a  head  of  320  feet. 
Kequired  the  diameter  of  the  suction,  and  delivery  pipes,  and  the  diameter  and 
number  of  the  rotors,  assuming  a  velocity  of  5 '5  feet  per  second  in  the  suction  and 
delivery  pipes,  and  a  manometric  efficiency  at  the  given  delivery  of  50  per  cent. 

Assume  provisionally  that  the  diameter  of  the  boss  of  the  wheel  is  3  inches. 

Let  d  be  the  external  diameter  of  the  annular  opening,  Fig.  295. 

Then,  l(#-V}         4. 

144       =  60x5-5' 
from  which  d  =  Q  inches  nearly. 

Taking  the  external  diameter  D  of  the  wheel  as  2d,  D  is  1  foot. 

Then,  vl  =  —^—  x  w  =  76  feet  per  sec. 

uo 

Assuming  radial  blades  at  outlet  the  head  lifted  by  each  wheel  is 

h  =  0-5.  ^  feet 

=  90  feet. 
Four  wheels  would  therefore  be  required. 

241.     Advantages  of  centrifugal  pumps. 

There  are  several  advantages  possessed  by  centrifugal  pumps. 

In  the  first  place,  as  there  are  no  sliding  parts,  such  as  occur  in 
reciprocating  pumps,  dirty  water  and  even  water  containing  com- 
paratively large  floating  bodies  can  be  pumped  without  greatly 
endangering  the  pump. 

Another  advantage  is  that  as  delivery  from  the  wheel  is 
constant,  there  is  no  fluctuation  of  speed  of  the  water  in  the 
suction  or  delivery  pipes,  and  consequently  there  is  no  necessity 
for  air  vessels  such  as  are  required  on  the  suction  and  delivery 
pipes  of  reciprocating  pumps.  There  is  also  considerably  less 
danger  of  large  stress  being  engendered  in  the  pipe  lines  by 
"water  hammer*." 

Another  advantage  is  the  impossibility  of  the  pressure  in  the 
*  See  page  384. 

28—2 


436  HYDRAULICS 

pump  casing  rising  above  that  of  the  maximum  head  which  the 
rotor  is  capable  of  impressing  upon  the  water.  If  the  delivery 
is  closed  the  wheel  will  rotate  without  any  danger  of  the  pressure 
in  the  casing  becoming  greater  than  the  centrifugal  head  (page 
335).  This  may  be  of  use  in  those  cases  where  a  pump  is  de- 
livering into  a  reservoir  or  pumping  from  a  reservoir.  In  the  first 
case  a  float  valve  may  be  fitted,  which,  when  the  water  rises  to 
a  particular  height  in  the  reservoir,  closes  the  delivery.  The 
pump  wheel  will  continue  to  rotate  but  without  delivering  water, 
and  if  the  wheel  is  running  at  such  a  velocity  that  the  centri- 
fugal head  is  greater  than  the  head  in  the  pipe  line  it  will  start 
delivery  when  the  valve  is  opened.  In  the  second  case  a  similar 
valve  may  be  used  to  stop  the  flow  when  the  water  falls  below  a 
certain  level.  This  arrangement  although  convenient  is  uneco- 
nomical, as  although  the  pump  is  doing  no  effective  work,  the 
power  required  to  drive  the  pump  may  be  more  than  50  per  cent, 
of  that  required  when  the  pump  is  giving  maximum  discharge. 

It  follows  that  a  centrifugal  pump  may  be  made  to  deliver 
water  into  a  closed  pipe  system  from  which  water  may  be  taken 
regularly,  or  at  intervals,  while  the  pump  continues  to  rotate  at  a 
constant  velocity. 

Pump  delivering  into  a  long  pipe  line.  When  a  centrifugal 
pump  or  air  fan  is  delivering  into  a  long  pipe  line  the  resistances 
will  vary  approximately  as  the  square  of  the  quantity  of  water 
delivered  by  the  pump. 

Let  p>2  be  the  absolute  pressure  per  square  inch  which  has 
to  be  maintained  at  the  end  of  the  pipe  line,  and  let  the 
resistances  vary  as  the  square  of  the  velocity  v  along  the  pipe. 
Then  if  the  resistances  are  equivalent  to  a  head  hf  =  kv2,  the 

pressure  head  —  at  the  pump  end  of  the  delivery  pipe  must  be 

w      w 

=P*   frQ2 

w  +  A2  ' 
A  being  the  sectional  area  of  the  pipe. 

/v\ 

Let  —  be  the  pressure  head  at  the  top  of  the  suction  pipe,  then 
the  gross  lift  of  the  pump  is 

r  _Pl  __P   _Pz        &Q2        P 

w      w     w      A2      w ' 
If,  therefore,  a  curve,  Fig.  298,  be  plotted  having 

fo-p)  ,  feQ2 

w        ~  A2 


CENTRIFUGAL   PUMPS 


437 


as  ordinates,  and  Q  as  abscissae,  it  will  be  a  parabola.  If  on 
the  same  figure  a  curve  having  h  as  ordinates  and  Q  as  abscissae 
be  drawn  for  any  given  speed,  the  intersection  of  these  two 
curves  at  the  point  P  will  give  the  maximum  discharge  the  pump 
will  deliver  along  the  pipe  at  the  given  speed. 


iris  C.  Ft.  per  Second/, 
Fig.  298. 

242.    Parallel  flow  turbine  pump. 

By  reversing  the  parallel  flow  turbine  a  pump  is  obtained 
which  is  similar  in  some  respects  to  the  centrifugal  pump,  but 
differs  from  it  in  an  essential  feature,  that  no  head  is  impressed  on 
the  water  by  centrifugal  forces  between  inlet  and  outlet.  It 
therefore  cannot  be  called  a  centrifugal  pump. 

The  vanes  of  such  a  pump  might  be  arranged  as  in  Fig.  299, 
the  triangles  of  velocities  for  inlet  and  outlet  being  as  shown. 

The  discharge  may  be  allowed  to  take  place  into  guide 
passages  above  or  below  the  wheel,  where  the  velocity  can  be 
gradually  reduced. 

Since  there  is  no  centrifugal  head  impressed  on  the  water 
between  inlet  and  outlet,  Bernoulli's  equation  is 

£.+*'=£  +^'. 

w     2g     w      2gr 

From  which,  as  in  the  centrifugal  pump, 


-  _ 

g        w      w     2g     2g' 

If  the  wheel  has  parallel  sides  as  in  Fig.  299,  the  axial  velocity 
of  flow  will  be  constant  and  if  the  angles  <£  and  0  are  properly 
chosen,  Vr  and  vr  may  be  equal,  in  which  case  the  pressure  at 
inlet  and  outlet  of  the  wheel  will  be  equal.  This  would  have 
the  advantage  of  stopping  the  tendency  for  leakage  through  the 
clearance  between  the  wheel  and  casing. 


438 


HYDRAULICS 


Such  a  pump  is  similar  to  a  reversed  impulse  turbine,  the 
guide  passages  of  which  are  kept  full.  The  velocity  with  which 
the  water  leaves  the  wheel  would  however  be  great  and  the  lift 
above  the  pump  would  depend  upon  the  percentage  of  the  velocity 
head  that  could  be  converted  into  pressure  head. 


Fig.  299. 

Since  there  is  no  centrifugal  head  impressed  upon  the  water, 
the  parallel-flow  pump  cannot  commence  discharging  unless  the 
water  in  the  pump  is  first  set  in  motion  by  some  external  means, 
but  as  soon  as  the  flow  is  commenced  through  the  wheel,  the  full 
discharge  under  full  head  can  be  obtained. 


Fig.  300. 


Fig.  301. 


To  commence  the  discharge,  the  pump  would  generally  have  to 
be  placed  below  the  level  of  the  water  to  be  lifted,  an  auxiliary 
lischarge  pipe  being  fitted  with  a  discharging  valve,  and  a  non- 
return valve  in  the  discharge  pipe,  arranged  as  in  Fig.  300. 


CENTRIFUGAL   PUMPS  439 

The  pump  could  be  started  when  placed  at  a  height  h0  above 
the  water  in  the  sump,  by  using  an  ejector  or  air  pump  to  exhaust 
the  air  from  the  discharge  chamber,  and  thus  start  the  flow 
through  the  wheel. 

243.     Inward  flow  turbine  pump. 

Like  the  parallel  flow  pump,  an  inward  flow  pump  if  constructed 
could  not  start  pumping  unless  the  water  in  the  wheel  were  first 
set  in  motion.  If  the  wheel  is  started  with  the  water  at  rest 
the  centrifugal  head  will  tend  to  cause  the  flow  to  take  place 
outwards,  but  if  flow  can  be  commenced  and  the  vanes  are 
properly  designed,  the  wheel  can  be  made  to  deliver  water  at  its 
inner  periphery.  As  in  the  centrifugal  and  parallel  flow  pumps, 
if  the  water  enters  the  wheel  radially,  the  total  lift  is 


TT  _     ii  =2i_       ,_  m 

g        w      w+2g     2g  ..................  U;" 

From  the  equation 

^VL2=pitv^_i^ 
w  +  2g      w*2g+2g     2g> 
it  will  be  seen  that  unless  Vr2  is  greater  than 

vl     tf__v? 
2g+2g     2g> 

IP 

P!  is  less  than  p,  and  ~-  will  then  be  greater  than   the  total 

lift  H. 

Very  special  precautions  must  therefore  be  made  to  diminish 
the  velocity  U  gradually,  or  otherwise  the  efficiency  of  the  pump 
will  be  very  low. 

The  centrifugal  head  can  be  made  small  by  making  the 
difference  of  the  inner  and  outer  radii  small. 

«  ^  +  ^.^i 

2g+2g     2g 

V  2 
is  made  equal  to  -~  ,  the  pressure  at  inlet  and  outlet  will  be  the 

same,  and  if  the  wheel  passages  are  carefully  designed,  the 
pressure  throughout  the  wheel  may  be  kept  constant,  and  the 
pump  becomes  practically  an  impulse  pump. 

There  seems  no  advantage  to  be  obtained  by  using  either 
a  parallel  flow  pump  or  inward  flow  pump  in  place  of  the  centri- 
fugal pump,  and  as  already  suggested  there  are  distinct  dis- 
advantages. 

244.     Reciprocating  pumps. 

A  simple  form  of  reciprocating  force  pump  is  shown  dia- 
grammatically  in  Fig.  301.  It  consists  of  a  plunger  P  working  in 


440 


HYDRAULICS 


Fig.  301  a.     Vertical  Single-acting  Reciprocating  Pump. 


RECIPROCATING   PUMPS  441 

a  cylinder  C  and  has  two  valves  Ys  and  YD,  known  as  the  suction 
and  delivery  valves  respectively.  A  section  of  an  actual  pump 
is  shown  in  Fig.  301  a. 

Assume  for  simplicity  the  pump  to  be  horizontal,  with  the 
centre  of  the  barrel  at  a  distance  h  from  the  level  of  the  water 
in  the  well;  h  may  be  negative  or  positive  according  as  the 
pump  is  above  or  below  the  surface  of  the  water  in  the  well. 

Let  B  be  the  height  of  the  barometer  in  inches  of  mercury. 
The  equivalent  head  H,  in  feet  of  water,  is 

13-596.  B     V1 
12        =  l  lddB> 

which  may  be  called  the  barometric  height  in  feet  of  water. 

When  B  is  30  inches  H  is  34  feet. 

When  the  plunger  is  at  rest,  the  valve  YD  is  closed  by  the  head 
of  water  above  it,  and  the  water  in  the  suction  pipe  is  sustained  by 
the  atmospheric  pressure. 

Let  h0  be  the  pressure  head  in  the  cylinder,  then 

h0  =  H-h, 
or  the  pressure  in  pounds  per  square  inch  in  the  cylinder  is 

P--43KH-W, 

p  cannot  become  less  than  the  vapour  tension  of  the  water.  At 
ordinary  temperatures  this  is  nearly  zero,  and  hn  cannot  be  greater 
than  34  feet. 

If  now  the  plunger  is  moved  outwards,  very  slowly,  and  there 
is  no  air  leakage  the  valve  Ys  opens,  and  the  atmospheric  pressure 
causes  water  to  rise  up  the  suction  pipe  and  into  the  cylinder, 
h0  remaining  practically  constant. 

On  the  motion  of  the  plunger  being  reversed,  the  valve  Ys 
closes,  and  the  water  is  forced  through  YD  into  the  delivery 
pipe. 

In  actual  pumps  if  h0  is  less  than  from  4  to  9  feet  the 
dissolved  gases  that  are  in  the  water  are  liberated,  and  it  is  there- 
fore practically  impossible  to  raise  water  more  than  from  25  to 
30  feet. 

Let  A  be  the  area  of  the  plunger  in  square  inches  and  L  the 
stroke  in  feet.  The  pressure  on  the  end  of  the  plunger  outside  the 
cylinder  is  equal  to  the  atmospheric  pressure,  and  neglecting 
the  friction  between  the  plunger  and  the  cylinder,  the  force  neces- 
sary to  move  the  plunger  is 

P  -  '43  {H  -  (H  -  h)}  A  =  -43/i .  A  Ibs., 
and  the  work  done  by  the  plunger  per  stroke  is 
E  =  -437i .  A  .  L  ft.  Ibs. 


442  HYDRAULICS 

If  Y  is  the  volume  displacement  per  stroke  of  the   plunger 

in  cubic  feet 

E  =  62'4/t .  V  ft.  Ibs. 

The  weight  of  water  lifted  per  stroke  is  '43AL  Ibs.,  and  the 
work  done  per  pound  is,  therefore,  h  foot  pounds. 

Let  Z  be  the  head  in  the  delivery  pipe  above  the  centre  of  the 
pump,  and  ud  the  velocity  with  which  the  water  leaves  the  delivery 
pipe. 

Neglecting  friction,  the  work  done  by  the  plunger  during  the 

2 

delivery  stroke  is  Z  +  ^  foot  pounds  per  pound,  and  the  total  work 
*9 

in  the  two  strokes  is  therefore  h  +  Z  +  --•  foot  pounds  per  pound. 

The  actual  work  done  on  the  plunger  will  be  greater  than  this 
due  to  mechanical  friction  in  the  pump,  and  the  frictional  and 
other  hydraulic  losses  in  the  suction  and  delivery  pipes,  and  at  the 
valves;  and  the  volume  of  water  lifted  per  suction  stroke  will 
generally  be  slightly  less  than  the  volume  moved  through  by  the 
plunger. 

Let  W  be  the  weight  of  water  lifted  per  minute,  and  ht  the 
total  height  through  which  the  water  is  lifted. 

The  effective  work  done  by  the  pump  is  W .  ht  foot  pounds  per 
minute,  and  the  effective  horse-power  is 

HP  = 


33,000 ' 

245.     Coefficient  of  discharge  of  the  pump.     Slip. 

The  theoretical  discharge  of  a  plunger  pump  is  the  volume 
displaced  by  the  plunger  per  stroke  multiplied  by  the  number  of 
delivery  strokes  per  minute. 

The  actual  discharge  may  be  greater  or  less  than  this  amount. 
The  ratio  of  the  discharge  per  stroke  to  the  volume  displaced  by 
the  plunger  per  stroke  is  the  Coefficient  of  discharge,  and  the 
difference  between  these  quantities  is  called  the  Slip. 

If  the  actual  discharge  is  less  than  the  theoretical  the  slip  is 
said  to  be  positive,  and  if  greater,  negative. 

Positive  slip  is  due  to  leakage  past  the  valves  and  plunger, 
and  in  a  steady  working  pump,  with  valves  in  proper  condition, 
should  be  less  than  five  per  cent. 

The  causes  of  negative  slip  and  the  conditions  under  which  it 
takes  place  will  be  discussed  later*. 

*  See  page  461. 


RECIPROCATING   PUMPS 


443 


246.     Diagram  of  work  done  by  the  pump. 

Theoretical  Diagram.  Let  a  diagram  be  drawn,  Fig.  302,  the 
ordinates  representing  the  pressure  in  the  cylinder  and  the  abscissae 
the  corresponding  volume  displacements  of  the  plunger.  The 
volumes  will  clearly  be  proportional  to  the  displacement  of  the 
plunger  from  the  end  of  its  stroke.  During  the  suction  stroke, 
on  the  assumption  made  above  that  the  plunger  moves  very 
slowly  and  that  therefore  all  frictional  resistances,  and  also  the 
inertia  forces,  may  be  neglected,  the  absolute  pressure  behind  the 
plunger  is  constant  and  equal  to  H  -  h  feet  of  water,  or  62'4  (H  -  h) 
pounds  per  square  foot,  and  on  the  delivery  stroke  the  pressure  is 

/  u  2x 

62*4  (  Z  +  H  +  ~—  J  pounds  per  square  foot. 

The  effective  work  done  per  suction  stroke  is  ABCD  which  equals 
62'4 .  h .  V,  and  during  the  delivery  stroke  is  EADF  which  equals 

62v 

and  EBCF  is  the  work  done  per  cycle,  that  is,  during  one  suction 
and  one  delivery  stroke. 

E  F 


so 

T 

40 

^JC) 

if|* 

20 

z 

.   20- 
Atrrv 

A                      ]            D 

Pressure, 

B 

T                                r^ 

c 

'                 T<-p£ 

0 

Fig.  302.     Theoretical  diagram  of  pressure  in  a  Reciprocating  Pump. 


59   Strokes  per  nvuvuuU, 


Fig.  303. 

Actual  diagram.  Fig.  303  shows  an  actual  diagram  taken  by 
means  of  an  indicator  from  a  single  acting  pump,  when  running 
at  a  slow  speed. 

The  diagram  approximates  to  the  rectangular  form  and  only 


444  HYDRAULICS 

differs  from  the  above  in  that  at  any  point  p  in  the  suction  stroke, 
pq  in  feet  of  water  is  equal  to  h  plus  the  losses  in  the  suction 
pipe,  including  loss  at  the  valve,  plus  the  head  required  to 
accelerate  the  water  in  the  suction  pipe,  and  qr  is  the  head 
required  to  lift  the  water  and  overcome  all  losses,  and  to  accelerate 
the  water  in  the  delivery  pipe.  The  velocity  of  the  plunger  being 
small,  these  correcting  quantities  are  practically  inappreciable. 

The  area  of  this  diagram  represents  the  actual  work  done  on 
the  water  per  cycle,  and  is  equal  to  W  (Z  +  h),  together  with  the 
head  due  to  velocity  of  discharge  and  all  losses  of  energy  in  the 
suction  and  delivery  pipes. 

It  will  be  seen  later  that  although  at  any  instant  the  pressure 
in  the  cylinder  is  effected  by  the  inertia  forces,  the  total  work 
done  in  accelerating  the  water  is  zero. 

247.  The  accelerations  of  the  pump  plunger  and  of  the 
water  in  the  suction  pipe. 

The  theoretical  diagram,  Fig.  302,  has  been  drawn  on  the 
assumption  that  the  velocity  of  the  plunger  is  very  small  and 
without  reference  to  the  variation  of  the  velocity  and  of  the 
acceleration  of  the  plunger,  but  it  is  now  necessary  to  consider 
this  variation  and  its  effect  on  the  motion  of  the  water  in  the  suction 
and  delivery  pipes.  To  realise  how  the  velocity  and  acceleration 
of  the  plunger  vary,  suppose  it  to  be  driven  by  a  crank  and 
connecting  rod,  as  in  Fig.  304,  and  suppose  the  crank  rotates  with 
a  uniform  angular  velocity  of  w  radians  per  second. 


Fig.  304. 

If  r  is  the  radius  of  the  crank  in  feet,  the  velocity  of  the  crank 
pin  is  V  =  o>r  feet  per  second.  For  any  crank  position  OC,  it  is 
proved  in  books  on  mechanism,  that  the  velocity  of  the  point  B  is 

01TC'     By  making   BD   equal  to^K  a  diagram  of  velocities 
EDF  is  found. 

When  CB  is  very  long  compared  with  CO,  OK  is  equal  to 
OC  sin  6,  and  the  velocity  v  of  the  plunger  is  then  V  sin  0,  and 


RECIPROCATING   PUMPS 


445 


EDF  is  a  semicircle.  The  plunger  then  moves  with  simple 
harmonic  motion. 

If  now  the  suction  pipe  is  as  in  Fig.  300,  and  there  is  to  be 
continuity  in  the  column  of  water  in  the  pipe  and  cylinder,  the 
velocity  of  the  water  in  the  pipe  must  vary  with  the  velocity  of 
the  plunger. 

Let  v  be  the  velocity  of  the  plunger  at  any  instant,  A  and 
a  the  cross-sectional  areas  of  the  plunger  and  of  the  pipe  respect- 
ively. Then  the  velocity  in  the  pipe  must  be  -  -  . 


As  the  velocity  of  the  plunger  is  continuously  changing,  it  is 
continuously  being  accelerated,  either  positively  or  negatively. 

Let  Z  be  the  length  of  the  connecting  rod  in  feet.  The 
acceleration*  F  of  the  point  B  in  Fig.  305,  for  any  crank  angle 
0,  is  approximately 


Plotting  F  as  BGr,  Fig.  305,  a  curve  of  accelerations  MNQ  is 
obtained. 

When  the  connecting  rod  is  very  long  compared  with  the 
length  of  the  crank,  the  motion  is  simple  harmonic,  and  the 
acceleration  becomes 

F  =  wV  cos  0, 

and  the  diagram  of  accelerations  is  then  a  straight  line. 

Velocity  and  acceleration  of  the  water  in  the  suction  pipe.  The 
velocity  and  acceleration  of  the  plunger  being  v  and  F  respectively, 
for  continuity,  the  velocity  of  the  water  in  the  pipe  must  be 


v  —  and  the  acceleration 
a 


F.A 


*  See  Balancing  of  Engines,  W.  E.  Dalby. 


446  HYDRAULICS 

248.  The  effect  of  acceleration  of  the  plunger  on  the 
pressure  in  the  cylinder  during  the  suction  stroke. 

When  the  velocity  of  the  plunger  is  increasing,  F  is  positive, 
and  to  accelerate  the  water  in  the  suction  pipe  a  force  P  is 
required.  The  atmospheric  pressure  has,  therefore,  not  only  to 
lift  the  water  and  overcome  the  resistance  in  the  suction  pipe, 
but  it  has  also  to  provide  the  necessary  force  to  accelerate  the 
water,  and  the  pressure  in  the  cylinder  is  consequently  diminished. 

On  the  other  hand,  as  the  velocity  of  the  plunger  decreases, 
F  is  negative,  and  the  piston  has  to  exert  a  reaction  upon  the 
water  to  diminish  its  velocity,  or  the  pressure  on  the  plunger  is 
increased. 

Let  L  be  the  length  of  the  suction  pipe  in  feet,  a  its  cross- 
sectional  area  in  square  feet,  fa  the  acceleration  of  the  water  in 
the  pipe  at  any  instant  in  feet  per  second  per  second,  and  w  the 
weight  of  a  cubic  foot  of  water. 

Then  the  mass  of  water  in  the  pipe  to  be  accelerated  is  w  .  a  .  L 
pounds,  and  since  by  Newton's  second  law  of  motion 
accelerating  force  =  mass  x  acceleration, 
the  accelerating  force  required  is 

-PJ     w  .  a  .  L     ~  ,, 
-  ./«lbs. 

The  pressure  per  unit  area  is 

P     to.L    -  ,, 

-  =  _./albs., 

and  the  equivalent  head  of  water  is 

7,      L    f 
ll<*  =  -  •?«> 

\j 

A 


or  since  /«  = 


. 
g.a 

This  may  be  large  if  any  one  of  the  three  quantities,  L,  —  ,  or 

.  ^ 

F  is  large. 

Neglecting  friction    and    other    losses    the    pressure    in    the 
cylinder  is  now 

H-&-&., 
and  the  head  resisting  the  motion  of  the  piston  is  h  +  ha. 

249.    Pressure  in  the  cylinder  during  the  suction  stroke 
when  the  plunger  moves  with  simple  harmonic  motion. 

If  the  plunger  be  supposed  driven  by  a  crank  and  very  long 


RECIPROCATING   PUMPS 


447 


connecting  rod,  the  crank  rotating  uniformly  with  angular  velocity 
o>  radians  per  second,  for  any  crank  displacement  #, 


and 


, 
hn  = 


, 
.  cos  0. 


g .  a 
The  pressure  in  the  cylinder  is 

TT     7      L  Ao>V  cos  0 
Jti  —  fi — . 

ga 

When  0  is  zero,  cos  6  is  unity,  and  when  6  is  90  degrees,  cos  0 
is  zero.  For  values  of  G  between  90  and  180  degrees,  cos#  is 
negative. 

The  variation  of  the  pressure  in  the  cylinder  is  seen  in 
Fig.  306,  which  has  been  drawn  for  the  following  data. 


Fig.  306. 

Diameter  of  suction  pipe  3J  inches,  length  12  feet  6  inches. 
Diameter  of  plunger  4  inches,  length  of  stroke  7|  inches. 

Number  of  strokes  per  minute  136.  Height  of  the  centre  of 
the  pump  above  the  water  in  the  sump,  8  feet.  The  plunger  is 
assumed  to  have  simple  harmonic  motion. 

The  plunger,  since  its  motion  is  simple  harmonic,  may  be 
supposed  to  be  driven  by  a  crank  3|  inches  long,  making  68  revo- 
lutions per  minute,  and  a  very  long  connecting  rod. 

The  angular  velocity  of  the  crank  is 

27T.68     ,_- 
to  =  =71  radians  per  second. 

The  acceleration  at  the  ends  of  the  stroke  is 

F2     n,         *7*12  .,  A'QIO 
=  (y.7*—  t  L    X  U  OL& 

=  15*7  feet  per  sec.  per  sec., 

t-y»)'~116* 

,       12-5 . 157 . 1-63     1AJ 

and  fia  =         — ^5 —      —  =  10  teet. 

oZ 


448  HYDRAULICS 

The  pressure  in  the  cylinder  neglecting  the  water  in  the 
cylinder  at  the  beginning  of  the  stroke  is,  therefore, 

34  _  (10 +  8) -16  feet, 

and  at  the  end  it  is  34  -  8  +  10  =  36  feet.     That  is,  it  is  greater 
than  the  atmospheric  pressure. 

When  0  is  90  degrees,  cos#  is  zero,  and  ha  is  therefore  zero, 
and  when  0  is  greater  than  90  degrees,  cos  0  is  negative. 

The  area  AEDF  is  clearly  equal  to  GrADH,  and  the  work  done 
per  suction  stroke  is,  therefore,  not  altered  by  the  accelerating 
forces;  but  the  rate  at  which  the  plunger  is  working  at  various 
points  in  the  stroke  is  affected  by  them,  and  the  force  required  to 
move  the  plunger  may  be  very  much  increased. 

In  the  above  example,  for  instance,  the  force  necessary  to 
move  the  piston  at  the  commencement  of  the  stroke  has  been 
more  than  doubled  by  the  accelerating  force,  and  instead  of 
remaining  constant  and  equal  to  '43 . 8 .  A  during  the  stroke,  it 

varies  from 

P  =  '43(8  +  10)A 

to  P  = '43  (8 -10)  A. 

Air  vessels.  In  quick  running  pumps,  or  when  the  length 
of  the  pipe  is  long,  the  effects  of  these  accelerating  forces  tend  to 
become  serious,  not  only  in  causing  a  very  large  increase  in  the 
stresses  in  the  parts  of  the  pump,  but  as  will  be  shown  later,  under 
certain  circumstances  they  may  cause  separation  of  the  water  in 
the  pipe,  and  violent  hammer  actions  may  be  set  up.  To  reduce 
the  effects  of  the  accelerating  forces,  air  vessels  are  put  on  the 
suction  and  delivery  pipes,  Figs.  310  and  311. 

250.  Accelerating  forces  in  the  delivery  pipe  of  a  plunger 
pump  when  there  is  no  air  vessel. 

When  the  plunger  commences  its  return  stroke  it  has  not  only 
to  lift  the  water  against  the  head  in  the  delivery  pipe,  but,  if  no 
air  vessel  is  provided,  it  has  also  to  accelerate  the  water  in  the 
cylinder  and  the  delivery  pipe.  Let  D  be  the  diameter,  a*  the  area, 
and  Li  the  length  of  the  pipe.  Neglecting  the  water  in  the 
cylinder,  the  acceleration  head  when  the  acceleration  of  the  piston 
is  F,  is 

,       Lj.A.F 

ha  =  —  —  , 

00i 

and  neglecting  head  lost  by  friction  etc.,  and  the  water  in  the 
cylinder,  the  head  resisting  motion  is 

z +  *.+£. 

If  F  is  negative,  ha  is  also  negative. 


RECIPROCATING   PUMPS 


449 


When  the  plunger  moves  with  simple  harmonic  motion  the 
diagram  is  as  shown  in  Fig.  307,  which  is  drawn  for  the  same 
data  as  for  Fig.  306,  taking  Z  as  20  feet,  Ln  as  30  feet,  and  the 
diameter  D  as  3|  inches. 


Fig.  307. 

The  total  work  done  on  the  water  in  the  cylinder  is  NJKM, 
which  is  clearly  equal  to  HJKL.  If  the  atmospheric  pressure  is 
acting  on  the  outer  end  of  the  plunger,  as  in  Fig.  301,  the  nett 
work  done  on  the  plunger  will  be  SNRMT,  which  equals  HSTL. 

251.  Variation  of  pressure  in  the  cylinder  due  to  friction 
when  there  is  no  air  vessel. 

Head  lost  by  friction  in  the  suction  and  delivery  pipes.  If  v  is 
the  velocity  of  the  plunger  at  any  instant  during  the  suction 
stroke,  d  the  diameter,  and  a  the  area  of  the  suction  pipe,  the 
velocity  of  the  water  in  the  pipe,  when  there  is  no  air  vessel,  is 

0A 

—  ,  and  the  head  lost  by  friction  at  that  velocity  is 

=WA«L 

f  * 


Similarly,  if  ai,  D,  and  LI  are  the  area,  diameter  and  length 
respectively  of  the  delivery  pipe,  the  head  lost  by  friction,  when 
the  plunger  is  making  the  delivery  stroke  and  has  a  velocity  v,  is 


When  the  plunger  moves  with  simple  harmonic  motion, 

v  =  o>r  sin  0, 


and 


h,= 


4/AVr2sin20L 


2gda* 


L.  H. 


29 


450 


HYDRAULICS 


If  the  pump  makes  n  strokes  per  second,  or  the  number  of 
revolutions  of  the  crank  is  ~  per  second,  and  18  is  the  length  of 
the  stroke, 


and 


=  2r. 


Substituting  for  w  and  r, 


~          2gda? 

Plotting  values  of  hf  at  various  points  along  the  stroke,  the 
parabolic  curve  EMF,  Fig.  308,  is  obtained. 

When  &  is  90  degrees,  sin#  is  unity,  and  hf  is  a  maximum. 
The  mean  ordinate  of  the  parabola,  which  is  the  mean  frictional 
head,  is  then 

2  /AWU2 

3  2gda?      ' 


M 


Fig.  308. 

and  since  the  mean  frictional  head  is  equal  to  the  energy  lost  per 
pound  of  water,  the  work  done  per  stroke  by  friction  is 

—  foot  Ibs., 


all  dimensions  being  in  feet. 


Fig.  309. 
Let  Do  be  the  diameter  of  the  plunger  in  feet.     Then 


and 


d* 


RECIPROCATING   PUMPS  451 

Therefore,  work  done  by  friction  per  suction   stroke,   when 
there  is  no  air  vessel  on  the  suction  pipe,  is 


d5 

The  pressure  in  the  cylinder  for  any  position  of  the  plunger 
during  the  suction  stroke  is  now,  Fig.  309, 
h/Q  =  H  —  h  —  fia  —  hf. 

At  the  ends  of  the  stroke  h/  is  zero,  and  for  simple  harmonic 
motion  ha  is  zero  at  the  middle  of  the  stroke. 

The  work  done  per  suction  stroke  is  equal  to  the  area 
AEMFD,  which  equals 

ARSD  +  EMF  =  ' 


Similarly,  during  the  delivery  stroke  the  work  done  is 


> 

The  friction  diagram  is  HKGr,  Fig.  309,  and  the  resultant 
diagram  of  total  work  done  during  the  two  strokes  is  EMFGKH. 

252.     Air  vessel  on  the  suction  pipe. 

As  remarked  above,  in  quick  running  pumps,  or  when  the 
lengths  of  the  pipes  are  long,  the  effects  of  the  accelerating  forces 
become  serious,  and  air  vessels  are  put  on  the  suction  and  delivery 
pipes,  as  shown  in  Figs.  310  and  311.  By  this  means  the  velocity 
in  the  part  of  the  suction  pipe  between  the  well  and  the  air 
vessel  is  practically  kept  constant,  the  water,  which  has  its 
velocity  continually  changing  as  the  velocity  of  the  piston 
changes,  being  practically  confined  to  the  water  in  the  pipe 
between  the  air  vessel  and  the  cylinder.  The  head  required  to 
accelerate  the  water  at  any  instant  is  consequently  diminished, 
and  the  friction  head  also  remains  nearly  constant. 

Let  /!  be  the  length  of  the  pipe  between  the  air  vessel  and 
the  cylinder,  I  the  length  from  the  well  to  the  air  vessel,  a  the 
cross-sectional  area  of  each  of  the  pipes  and  d  the  diameter  of  the 
pipe. 

Let  hv  be  the  pressure  head  in  the  air  vessel  and  let  the  air 
vessel  be  of  such  a  size  that  the  variation  of  the  pressure  may  for 
simplicity  be  assumed  negligible. 

Suppose  now  that  water  flows  from  the  well  up  the  pipe  AB 
continuously  and  at  a  uniform  velocity.  The  pump  being  single 
acting,  while  the  crank  makes  one  revolution,  the  quantity  of 
water  which  flows  along  AB  must  be  equal  to  the  volume  the 
plunger  displaces  per  stroke. 

29—2 


452 


HYDRAULICS 


The  time  for  the  crank  to  make  one  revolution  is 


t  =  —  sees., 

U) 


therefore,  the  mean  velocity  of  flow  is 

^A^rw^  Ao>r 

/  2Awr  \ 

(For  a  double  acting  pump  vm  =  —  —  .  J 

During  the  delivery  stroke,  all  the  water  is  entering  the  air 
vessel,  the  water  in  the  pipe  BC  being  at  rest. 


Fig.  310. 

Then  by  Bernoulli's  theorem,  including  friction  and  the  velocity 
head,  other  losses  being  neglected,  the  atmospheric  head 

A2  o>V     4/AWZ  n, 

K  =  x  +  hv  +  n  —  2—  r+  ^    ,  a  2  ...............  (1). 

2ga    7T2  2 


The  third  and  fourth  quantities  of  the  right-hand  part  of  the 
equation  will  generally  be  very  small  and  hv  is  practically  equal 
to  H-». 

When  the  suction  stroke  is  taking  place,  the  water  in  the  pipe 
BC  has  to  be  accelerated. 

Let  HB  be  the  pressure  head  at  the  point  B,  when  the  velocity 
of  the  plunger  is  v  feet  per  second,  and  the  acceleration  F  feet  per 
second  per  second. 


RECIPROCATING   PUMPS  453 

Let  hf  be  the  loss  of  head  by  friction  in  AB,  and  In,}  the  loss  in 

BC.    The  velocity  of  flow  along  BC  is  — ,  and  the  velocity  of 

a 

flow  from  the  air  vessel  is,  therefore, 

v.A 


a         ira 

Then  considering  the  pipe  AB, 

AW 

HB~       ~Wtf~  " 

and  from  consideration  of  the  pressures  above  B, 


HR  — 


Neglecting  losses  at  the  valve,  the  pressure  in  the  cylinder  is 
then  approximately 

%       TT       1'     AZlF 

ft,=H,-*,-  — 

„     ,     AW    ,      ,,    AZ.F 
-h- 


Neglecting  the  small  quantity  ^  —  ^—z  , 

/to  =  H  -  h  -  (hf+  hf')  -  A^lF  . 

For  a  plunger  moving  with  simple  harmonic  motion 

TT     i.     4/a>V2AV^      7    •  2/j\     A^<o2rcos<9 
h0  =  H  -  h  -  ^  —  o~    ~2  +  Zi  sm2  ^    -  -  . 

2ga?d    \7r2  Jag 

By  putting  the  air  vessel  near  to  the  cylinder,  thus  making 
li  small,  the  acceleration  head  becomes  very  small  and 

ho  =  H-h-hf  nearly, 
and  for  simple  harmonic  motion 

^°= 

The  mean  velocity  in  the  suction  pipe  can  very  readily  be 
determined  as  follows. 

Let  Q  be  the  quantity  of  water  lifted  per  second  in  cubic  feet. 
Then  since  the  velocity  along  the  suction  pipe  is  practically 

constant  vm  =  —  and  the  friction  head  is 


454 


HYDKAULICS 


When  the  pump  is  single  acting  and  there  are  n  strokes  per 
second, 


K.ls.n 
and  therefore, 

and 

If  the  pump  is  double  acting, 

h  =  2fAWl 

For  the  same  length  of  suction  pipe  the  mean  friction  head, 
when  there  is  no  air  vessel  and  the  pump  is  single  acting,  is  \T? 
times  the  friction  head  when  there  is  an  air  vessel. 

253.    Air  vessel  on  the  delivery  pipe. 

An  air  vessel  on  the  delivery  pipe  serves  the  same  purpose 
as  on  the  suction  pipe,  in  diminishing  the  mass  of  water  which 
changes  its  velocity  as  the  piston  velocity  changes. 


Fig.  311. 

As  the  delivery  pipe  is  generally  much  longer  than  the  suction 
pipe,  the  changes  in  pressure  due  to  acceleration  may  be  much 
greater,  and  it  accordingly  becomes  increasingly  desirable  to 
provide  an  air  vessel. 

Assume  the  air  vessel  so  large  that  the  pressure  head  in  it 
remains  practically  constant. 


RECIPROCATING   PUMPS  455 

Let  Z2,  Fig.  311,  be  the  length  of  the  pipe  between  the  pump  and 
the  air  vessel,  ld  be  the  length  of  the  whole  pipe,  and  &i  and  D  the 
area  and  diameter  respectively  of  the  pipe. 

Let  7^2  be  the  height  of  the  surface  of  the  water  in  the  air  vessel 
above  the  centre  of  the  pipe  at  B,  and  let  Hr  be  the  pressure  head 
in  the  air  vessel.  On  the  assumption  that  Hr  remains  constant, 
the  velocity  in  the  part  BC  of  the  pipe  is  practically  constant. 

Let  Q  be  the  quantity  of  water  delivered  per  second. 

The  mean  velocity  in  the  part  BC  of  the  delivery  pipe  will  be 


.    .  .   . 

The  friction  head  in  this  part  of  the  pipe  is  constant  and  equal  to 


Considering  then  the  part  BC  of  the  delivery  pipe,  the  total 
head  at  B  required  to  force  the  water  along  the  pipe  will  be 


But  the  head  at  B  must  be  equal  to  Hv  +  h2  nearly,  therefore, 

-W  +  H     ...............  (1). 


In  the  part  AB  of  the  pipe  the  velocity  of  the  water  will  vary 
with  the  velocity  of  the  plunger. 

Let  v  and  F  be  the  velocity  and  acceleration  of  the  plunger 
respectively. 

Neglecting  the  water  in  the  cylinder,  the  head  Hr  resisting  the 
motion  of  the  plunger  will  be  the  head  at  B,  plus  the  head 
necessary  to  overcome  friction  in  AB,  and  to  accelerate  the  water 
in  AB. 

™      *  TT      T,      4/Z2^2A2     F.A.Z2 

Therefore,     Hr  =  H,  +  h> 


For  the  same  total  length  of  the  delivery  pipe  the  acceleration 
head  is  clearly  much  smaller  than  when  there  is  no  air  vessel. 
Substituting  for  Hr  +  h»  from  (1), 

r.A.fc 


If  the  pump  is  single  acting  and  the  plunger  moves  with  simple 
harmonic  motion  and  makes  n  strokes  per  second, 


Am 
and  u  =  -- 


456  HYDRAULICS 

Therefore. 


Neglecting  the  friction  head  in  Z2  and  assuming  12  small  com 
pared  with  ld, 


254.     Separation  during  the  suction  stroke. 

In  reciprocating  pumps  it  is  of  considerable  importance  that 
during  the  stroke  no  discontinuity  of  flow  shall  take  place,  or 
in  other  words,  no  part  of  the  water  in  the  pipe  shall  separate 
from  the  remainder,  or  from  the  water  in  the  cylinder  of  the  pump. 
Such  separation  causes  excessive  shocks  in  the  working  parts  of 
the  pump  and  tends  to  broken  joints  and  pipes,  due  to  the  hammer 
action  caused  by  the  sudden  change  of  momentum  of  a  large  mass 
of  moving  water  overtaking  the  part  from  which  it  has  become 
separated. 

Consider  a  section  AB  of  the  pipe,  Fig.  301,  near  to  the  inlet 
valve.  For  simplicity,  neglect  the  acceleration  of  the  water  in  the 
cylinder  or  suppose  it  to  move  with  the  plunger,  and  let  the 
acceleration  of  the  plunger  be  F  feet  per  second  per  second. 

If  now  the  water  in  the  pipe  is  not  to  be  separated  from  that  in 
the  cylinder,  the  acceleration  fa  of  the  water  in  the  pipe  must  not 

FA 

be  less  than feet  per  second  per  second,  or  separation  will  not 

a 

FA 
take  place  as  long  as  -  -  </a. 

-HI    A 

If  fa  at  any  instant  becomes  equal  to ,  and  fa  is  not  to  be- 

CL 

FA 

come  less  than  —  ,  the  diminution  d/of /a,  when  F  is  diminished 

CL 

by  a  small  amount  dF,  must  not  be  less  than  —  oF,  or  in  general 

CL 

the  differential  of  fa  must  not  be  less  than  —  times  the  differential 

a 
of  F. 

The  general  condition  for  no  separation  is,  therefore, 

Perhaps  a  simpler  way  to  look  at  the  question  is  as  follows. 

Let  it  be  supposed  that  for  given  data  the  curve  of  pressures 
in  the  cylinder  during  the  suction  stroke  has  been  drawn  as  in 
Fig.  309.  In  this  figure  the  pressure  in  the  cylinder  always  remains 
positive,  but  suppose  some  part  of  the  curve  of  pressures  EF  to 


RECIPROCATING   PUMPS 


457 


come  below  the  zero  line  BC  as  in  Fig.  312  *.  The  pressure  in  the 
cylinder  then  becomes  negative;  but  it  is  impossible  for  a  fluid 
to  be  in  tension  and  therefore  discontinuity  in  the  flow  must 
occur  t. 

In  actual  pumps  the  discontinuity  will  occur,  if  the  curve  EFGr 
falls  below  the  pressure  at  which  the  dissolved  gases  are  liberated, 
or  the  pressure  head  becomes  less  than  from  4  to  10  feet. 


B 


D 


Fig.  312. 

At  the  dead  centre  the  pressure  in  the  cylinder  just  becomes 
zero  when  h  +  ha  =  H,  and  will  become  negative  when  h  +  ha  >  H. 
Theoretically  for  no  separation  at  the  dead  centre,  therefore, 


-      or 


ga 


If  separation  takes  place  when  the  pressure  head  is  less  than 
some  head  hm,  for  no  separation, 

ha  =  H  —  hm  —  h, 

•ci 

and 


a    '  I 

Neglecting  the  water  in  the  cylinder,  at  any  other  point  in  the 
stroke,  the  pressure  is  negative  when 


^A*>H 
2atf  >H' 


~P  A     T 

That  is,  when         In  +  -       -  + 
a    g 


And  the  condition  for  no  separation,  therefore,  is 


FA 

a 


(2). 


*  See  also  Fig.  315,  page  459. 

t  Surface  tension  of  fluids  at  rest  is  not  alluded  to. 


458  HYDRAULICS 

255.     Separation    during   the   suction    stroke   when   the 
plunger  moves  with  simple  harmonic  motion. 

When  the  plunger  is  driven  by  a  crank  and  very  long  con- 
necting rod,  the  acceleration  for  any  crank  angle  0  is 

F  =  a,2  r  cos  0, 
or  if  the  pump  makes  n  single  strokes  per  second, 

(o=  Trn, 

o      o 

and  F  =  "rf-tf  .  r  cos  0=    ~  -  .  ls  cos  0, 

ls  being  the  length  of  the  stroke. 

F  is  a  maximum  when  0  is  zero,  and  separation  will  not  take 
place  at  the  end  of  the  stroke  if 

A    2    ^g(H.-hm-h) 
a  a  L 

and  will  just  not  take  place  when 

A  7T2    A    27         /H-frm- 

awV°r  2-awZ'  =  n—  L 
The  minimum  area  of  the  suction  pipe  for  no  separation  is, 
therefore, 

AwVL  xQv 

a=  —  Tff  —  7  --  y-r-  ........................  (o) 

gCBL-hm-h) 

and  the  maximum  number  of  single  strokes  per  second  is 


_ 

~ 


A.Z..L 

Separation  actually  takes  place  at  the  dead  centre  at  a  less 
number  of  strokes  than  given  by  formula  (4),  due  to  causes 
which  could  not  very  well  be  considered  in  deducing  the  formula. 

Example.  A  single  acting  pump  has  a  stroke  of  1\  inches  and  the  plunger  is 
4  inches  diameter.  The  diameter  of  the  suction  pipe  is  3^  inches,  the  length 
12-5  feet,  and  the  height  of  the  centre  of  the  pump  above  the  water  in  the  well  is 
8  feet. 

To  find  the  number  of  strokes  per  second  at  which  separation  will  take  place, 
assuming  it  to  do  so  when  the  pressure  head  falls  below  10  feet. 

H-/z  =  26feet, 

-=1-63, 
a 


and,  therefore,  „  =  1      /    64x26 

TT  V  1-63  x  7-5 


x!2 
5  x  12-5 


7T 


=  210  strokes  per  minute. 

Nearly  all  actual  diagrams  taken  from  pumps,  Figs.  313—315, 
have  the   corner   at  the   commencement   of    the   suction    stroke 


RECIPROCATING    PUMPS 


459 


rounded  off,  so  that  even  at  very  slow  speeds  slight  separation 
occurs.  The  two  principal  causes  of  this  are  probably  to  be  found 
first,  in  the  failure  of  the  valves  to  open  instantaneously,  and 
second,  in  the  elastic  yielding  of  the  air  compressed  in  the  water 
at  the  end  of  the  delivery  stroke. 


136  Strokes  per  Truuvuutt> 


Line 


Fig.  315. 

The  diagrams  Figs.  303  and  313 — 315,  taken  from  a  single-acting 
pump,  having  a  stroke  of  7J  inches,  and  a  ram  4  inches  diameter, 
illustrate  the  effect  of  the  rounding  of  the  corner  in  producing 
separation  at  a  less  speed  than  that  given  by  equation  (4). 

Even  at  59  strokes  per  minute,  Fig.  303,  at  the  dead  centre  a 
momentary  separation  appears  to  have  taken  place,  and  the  water 
has  then  overtaken  the  plunger,  the  hammer  action  producing 
vibration  of  the  indicator.  In  Figs.  313 — 315,  the  ordinates  to  the 
line  rs  give  the  theoretical  pressures  during  the  suction  stroke. 
The  actual  pressures  are  shown  by  the  diagram.  At  136  strokes 


460  HYDRAULICS 

per  minute  at  the  point  e  in  the  stroke  the  available  pressure  is 
clearly  less  than  ef  the  head  required  to  lift  the  water  and  to 
produce  acceleration,  and  the  water  lags  behind  the  plunger. 
This  condition  obtains  until  the  point  a  is  passed,  after  which 
the  water  is  accelerated  at  a  quicker  rate  than  the  piston,  and 
finally  overtakes  it  at  the  point  b,  when  it  strikes  the  plunger  and 
the  indicator  spring  receives  an  impulse  which  makes  the  wave 
form  on  the  diagram.  At  230  strokes  per  minute,  the  speed  being 
greater  than  that  given  by  the  formula  when  hm  is  assumed  to 
be  10  feet,  the  separation  is  very  pronounced,  and  the  water  does 
not  overtake  the  piston  until  '7  of  the  stroke  has  taken  place.  It 
is  interesting  to  endeavour  to  show  by  calculation  that  the  water 
should  overtake  the  plunger  at  b. 

While  the  piston  moves  from  a  to  b  the  crank  turns  through 
70  degrees,  in  TeT°T  .  •££;  seconds  =  '101  seconds.  Between  these  two 
points  the  pressure  in  the  cylinder  is  2  Ibs.  per  sq.  inch,  and 
therefore  the  head  available  to  lift  the  water,  to  overcome  all 
resistances  and  to  accelerate  the  water  in  the  pipe  is  29'3  feet. 

The  height  of  the  centre  of  the  pump  is  6'  3"  above  the  water 
in  the  sump.  The  total  length  of  the  suction  pipe  is  about 
12*5  feet,  and  its  diameter  is  3J  inches. 

Assuming  the  loss  of  head  at  the  valve  and  due  to  friction  etc., 
to  have  a  mean  value  of  2*5  feet,  the  mean  effective  head  accele- 
rating the  water  in  the  pipe  is  20'5  feet.  The  mean  acceleration 
is,  therefore, 

,      20-5  x  32     _  „  , 
ja  =  ~~<vc~~  =  52  5  feet  per  sec.  per  sec. 


When  the  piston  is  at  g  the  water  will  be  at  some  distance 
behind  the  piston.  Let  this  distance  be  z  inches  and  let  the 
velocity  of  the  water  be  u  feet  per  sec.  Then  in  the  time  it 
takes  the  crank  to  turn  through  70  degrees  the  water  will  move 
through  a  distance 

S  =  ut  +  %fat2 

=  O'lOlu  +  J52'5  x  -0102  feet 
-  l'2u  +  3'2  inches. 

The  horizontal  distance  ab  is  4'2  inches,  so  that  z  +  4*2  inches 
should  be  equal  to  l'2u  +  3'2  inches. 

The  distance  of  the  point  g  from  the  end  of  the  stroke  is 
'84  inch  and  the  time  taken  by  the  piston  to  move  from  rest  to  g, 
is  0*058  second.  The  mean  pressure  accelerating  the  water  during 
this  time  is  the  mean  ordinate  of  akm  when  plotted  on  a  time 
base;  this  is  about  5  Ibs.  per  sq.  inch,  and  the  equivalent  head  is 
12-8  feet. 


RECIPROCATING   PUMPS  461 

The  frictional  resistances,  which  vary  with  the  velocity,  will  be 
small.  Assuming  the  mean  frictional  head  to  be  '25  foot,  the  head 
causing  acceleration  is  12'55  feet  and  the  mean  acceleration  of  the 
water  in  the  pipe  while  the  piston  moves  from  rest  to  g  is, 
therefore, 

,     12-55  x  32 

fa  = 


The  velocity  in  the  pipe  at  the  end  of   0'058  second,  should 
therefore  be 

v  =  32  x  -058  -  1'86  feet  per  sec. 

and  the  velocity  in  the  cylinder 

T86     110- 
u  =  ^7^5  =  1  153  ieet  per  sec. 

1  Oo 

Since  the  water  in  the  pipe  starts  from  rest  the  distance  it 
should  move  in  0'058  second  is 

12.J32.('058)2=-65in., 

and  the  distance  it  should  advance  in  the  cylinder  is 

0-65  . 


so  that  z  is  0'4  in. 

Then  z  +  4'2  ins.  -  4'6, 

and  \'2u  +  3'2  ins.  =  4*57  ins. 

The  agreement  is,  therefore,  very  close,  and  the  assumptions 
made  are  apparently  justified. 

256.  Negative  slip  in  a  plunger  pump. 

Fig.  315  shows  very  clearly  the  momentary  increase  in  the 
pressure  due  to  the  blow,  when  the  water  overtakes  the  plunger, 
the  pressure  rising  above  the  delivery  pressure,  and  causing 
discharge  before  the  end  of  the  stroke  is  reached.  If  no  separa- 
tion had  taken  place,  the  suction  pressure  diagram  would  have 
approximated  to  the  line  rs  and  the  delivery  valve  would  still 
have  opened  before  the  end  of  the  stroke  was  reached. 

The  coefficient  of  discharge  is  T025,  whereas  at  59  strokes 
per  minute  it  is  only  0'975. 

257.  Separation  at  points  in  the  suction  stroke  other  than 
at  the  end  of  the  stroke. 

The  acceleration  of   the  plunger  for  a  crank  displacement  6 

is  cos  6,   and   therefore   for   no   separation   at   any  crank 

a 

angle  0 


462  HYDRAULICS 

Putting  in  the  value  of  h/9  and  differentiating  both  sides  of  the 
equation,  and  using  the  result  of  equation  (1),  page  456, 


-  o>V  sin  0^^-—Jl  +  sin  6  cos  6, 

a  L    or  \          d   J 

(       4f  L\ 
from  which  aL  £  A  f  1  +  ~j-  1  ^  cos  0. 

Separation  will  just  not  take  place  if 

aL  =  A.r(l  +  4£k)  cos  6, 

or  when  cos0  =  -  a     .  .....................  (2). 


Since  cos  0  cannot  be  greater  than  unity,  there  is  no  real 
solution  to  this  equation,  unless  Ar  (l  +  ^j-J  is  equal  to  or 
greater  than  aL 

If,  therefore,  3*r  is  supposed  equal  to  zero,  and  aL  the  volume 

of  the  suction  pipe  is  greater  than  half  the  volume  of  the  cylinder, 
separation  cannot  take  place  if  it  does  not  take  place  at  the  dead 
centre. 

In  actual  pumps,  aL  is  not  likely  to  be  less  than  Ar,  and 
consequently  it  is  only  necessary  to  consider  the  condition  for  no 
separation  at  the  dead  centre. 

258.     Separation  with  a  large  air  vessel  on  the  suction  pipe. 

To  find  whether  separation  will  take  place  with  a  large  air 
vessel  on  the  suction  pipe,  it  is  only  necessary  to  substitute  in 
equations  (2),  section  255,  and  (3),  (4),  section  256,  hv  of  Fig.  310 
for  H,  li  for  L,  and  hi  for  h.  In  Fig.  310,  hi  is  negative. 

For  no  separation  when  the  plunger  is  at  the  end  of  the  stroke 
the  minimum  area  of  the  pipe  between  the  air  vessel  and  the 
cylinder  is 


a  = 


(hv-hm-hi)  ' 

Substituting  for  hv  its  value  from  equation  (1),  section  253,  and 
h  torn -h,  Fig.  310, 


a  = 


If  the  velocity  and  friction   heads,  in  the   denominator,  be 
neglected  as  being  small  compared  with  (H  -  h),  then, 


g(H-hm-h)' 


RECIPROCATING   PUMPS 
The  maximum  number  of  strokes  is 


463 


A.  pump  can  therefore  be  run  at  a  much  greater  speed,  without 
fear  of  separation,  with  an  air  vessel  on  the  suction  pipe,  than 
without  one. 

259.     Separation  in  the  delivery  pipe. 

Consider  a  pipe  as  shown  in  Fig.  316,  the  centre  of  CD  being  at 
a  height  Z  above  the  centre  of  AB. 

Let  the  pressure  head  at  D  be  H0,  which,  when  the  pipe 
discharges  into  the  atmosphere,  becomes  H. 

Let  I,  li  and  Z2  be  the  lengths  of  AB,  BC  and  CD  respectively, 
hf,  hfl  and  hfz  the  losses  of  head  by  friction  in  these  pipes  when  the 
plunger  has  a  velocity  v,  and  hm  the  pressure  at  which  separation 
actually  takes  place. 


I  ~H 


Fig.  316. 

Suppose  now  the  velocity  of  the  plunger  is  diminishing,  and  its 
retardation  is  F  feet  per  second  per  second.     If  there  is  to  be 

F  A 

continuity,  the  water  in  the  pipe  must  be  also  retarded  by  - 

ft 

feet  per  second  per  second,  and   the   pressure  must  always  be 
positive  and  greater  than  hm. 

Let  Hc  be  the  pressure  at  C ;  then  the  head  due  to  acceleration 
in  the  pipe  DC  is 


and  if  the  pipe  CD  is  full  of  water 

w      FAk 

-tic  —  -tJ-o  — 

9 
which  becomes  negative  when 


4G4  HYDRAULICS 

The  condition  for  no  separation  at  C  is,  therefore, 
H.-JL.-A 

-H-O  ~  Um  —  /I/  r 

or  separation  takes  place  when 


At  the  point  B  separation  will  take  place  if 


a        g 
and  at  the  point  A  if 


At  the  dead  centre  i?  is  zero,  and  the  friction  head  vanishes. 
For  no  separation  at  the  point  C  it  is  then  necessary  that 


for  no  separation  at  B 

, 

±±0  +  ZJ  —  Alm 

ga 

and  for  no  separation  at  A 


For  given  values  of  H0,  F  and  Z,  the  greater  Z2,  the  more  likely 
is  separation  to  take  place  at  C,  and  it  is  therefore  better,  for 
a  given  total  length  of  the  discharge  pipe,  to  let  the  pipe  rise  near 
the  delivery  end,  as  shown  by  fesS"  lines,  rather  than  as  shown 
by  the  <rf«  lines. 

If  separation  does  not  take  place  at  A  it  clearly  will  not  take 
place  at  B. 

Example.  The  retardation  of  the  plunger  of  a  pump  at  the  end  of  its  stroke 
is  8  feet  per  second  per  second.  The  ratio  of  the  area  of  the  delivery  pipe  to  the 
plunger  is  2,  and  the  total  length  of  the  delivery  pipe  is  152  feet.  The  pipe  is 
horizontal  for  a  length  of  45  feet,  then  vertical  for  40  feet,  then  rises  5  feet  on 
a  slope  of  1  vertical  to  3  horizontal  and  is  then  horizontal,  and  discharges  into 
the  atmosphere.  Will  separation  take  place  on  the  assumption  that  the  pressure 
head  cannot  be  less  than  7  feet  ? 

Ans.     At  the  bottom  of  the  sloping  pipe  the  pressure  is 

The  pressure  head  is  therefore  less  than  7  feet  and  separation  will  take  place. 
The  student  should  also  find  whether  there  is  separation  at  any  other  point. 


RECIPROCATING   PUMPS  4G5 

260.  Diagram  of  pressure  in  the  cylinder  and  work  done 
during  the  suction  stroke,  considering  the  variable  quantity  of 
water  in  the  cylinder. 

It  is  instructive  to  consider  the  suction  stroke  a  little  more  in 
detail. 

Let  v  and  F  be  the  velocity  and  acceleration  respectively  of 
the  piston  at  any  point  in  the  stroke. 

As  the  piston  moves  forward,  water  will  enter  the  pipe  from  the 
well  and  its  velocity  will  therefore  be  increased  from  zero  to 

A 

v  .  —  -,  the  head  required  to  give  this  velocity  is 

& 


On  the  other  hand  water  that  enters  the  cylinder  from  the  pipe 
is  diminished  in  velocity  from  -  -  to  v,  and  neglecting  any  loss  due 

CL 

to  shock  or  due  to  contraction  at  the  valve  there  is  a  gain  of 
pressure  head  in  the  cylinder  equal  to 


-~ 

*~2ga?     2g 
The  friction  head  in  the  pipe  is 


The  head  required  to  accelerate  the  water  in  the  pipe  is 

FAL 


The   mass  of    water  to  be  accelerated  in   the   cylinder  is   a 
variable  quantity  and  will  depend  upon  the  plunger  displacement. 
Let  the  displacement  be  x  feet  from  the  end  of  the  stroke. 

/7M    A    SY> 

The  mass  of  water  in  the  cylinder  is  -     -  Ibs.  and  the  force 

g 

required  to  accelerate  it  is 


and  the  equivalent  head  is 

P 


g 


wA.        g 

The  total  acceleration  head  is  therefore 
F/       LAN 


/       LA\ 

(SB  + ). 

\         a  / 


L.  H.  30 


6  HYDRAULICS 

Now  let  Ho  be  the  pressure  head  in  the  cylinder,  then 

_  7,  _  j^!  A!     ^  A2  _  ^  _  4/LAV  _  F  /       LAN 
H°=      ~k~2g^+2g  a2     2g      2g.da2      g  V    '    a  j 


g  \         a 

When  the  plunger  moves  with  simple  harmonic  motion,  and  is 
driven  by  a  crank  of  radius  r  rotating  uniformly  with  angular 
velocity  w,  the  displacement  of  the  plunger  from  the  end  of  the 
stroke  is  r  (1  -  cos  0),  the  velocity  wr  sin  0  and  its  acceleration  is 
o>V  cos  0. 
Therefore 

oiV.Bm'0     4/LAV 

H°-  2g  2gda> 

L    „  A        ,     oVcosfl     eoVcos2^       ,-, 
--  w-r—  cos0  --  -+-       -    ...(6). 

9        a  9  9 

Work  done,  during  the  suction  stroke.  Assuming  atmospheric 
pressure  on  the  face  of  the  plunger,  the  pressure  per  square  foot 
resisting  its  motion  is 


For  any  small  plunger  displacement  dx,  the  work  done  is, 
therefore, 

A  (H  -  Ho)  w  .  dx, 

and  the  total  work  done  during  the  stroke  is 


o 
The  displacement  from  the  end  of  the  stroke  is 

x  =  r  (I  -cos0), 
;and  therefore  dx  =  r  sin  QdO, 

and  E  =  fw  .  A  (H  -  H0)  r  sin  OdO. 

Jo 

Substituting  for  H0  its  value  from  equation  (6) 

-^  f»       4/LAW2  sin2  6     o>Vsin20 

E  =  w  .  Ar      h  +  -*—     —  j—2  --  +  - 

7o  2gda2  2g 

wV  cos  0     <oV  cos2  0     L  A  /»!•/»  JA 

+  -  -  +  --  w2r  cos  6}  sm  OdO. 

9  9  9  a 

The  sum  of  the  integration  of  the  last  four  quantities  of  this 
expression  is  equal  to  zero,  so  that  the  work  done  by  the 
accelerating  forces  is  zero,  and 


Jo 


4/LA2coV2 


RECIPROCATING   PUMPS 


467 


Or  the  work  done  is  that  required  to  lift  the  water  through 
a  height  h  together  with  the  work  done  in  overcoming  the 
resistance  in  the  pipe. 

Diagrams  of  pressure  in  the  cylinder  and  of  work  done  per 
stroke.  The  resultant  pressure  in  the  cylinder,  and  the  head 
resisting  the  motion  of  the  piston  can  be  represented  diagram- 
matically,  by  plotting  curves  the  ordinates  of  which  are  equal  to 
Ho  and  H-H0  as  calculated  from  equations  (2)  and  (3).  For 
clearness  the  diagrams  corresponding  to  each  of  the  parts  of 
equation  (2)  are  drawn  in  Figs.  318 — 321  and  in  Fig.  317  is  shown 
the  combined  diagram,  any  ordinate  of  which  equals 


-^ 


Figs.  318,  319,  320.  Figs.  321,  322. 

In  Fig.  318  the  ordinate  cd  is  equal  to 


and  the  curve  HJK  is  a  parabola,  the  area  of  which  is 


30—2 


468  HYDRAULICS 

In  Fig.  319,  the  ordinate  e/is 


and  the  ordinate  gh  of  Fig.  320  is 

+  ^cos2*. 
9 

The  areas  of  the  curves  are  respectively 


2  toV  7         ,  1  eoV    7 

3  W  *          3^~      ' 

and  are  therefore  equal;  and  since  the  ordinates  are  always  of 
opposite  sign  the  sum  of  the  two  areas  is  zero. 
In  Fig.  322,  ~km  is  equal  to 

o>V2  cos  0 

~T~ 

and  Id  to 

o>V        a  /        L  .  A\ 

-  cos  0  1  x  +  -     -  )  . 
0  a    / 

Since  cos  0  is  negative  between  90°  and  180°  the  area  WXY  is 
equal  to  YZU. 

Fig.  321  has  for  its  ordinate  at  any  point  of  the  stroke,  the 
head  H-H0  resisting  the  motion  of  the  piston. 

This  equals  h  +  Id  +  cd  +  ef—  gh, 

and  the  curve  NPS  is  clearly  the  curve  GrFE,  inverted. 

The  area  VNST  measured  on  the  proper  scale,  is  the  work  done 
per  stroke,  and  is  equal  to  VMRT  +  HJK. 

The  scale  of  the  diagram  can  be  determined  as  follows. 

Since  h  feet  of  water  =  62'4/z,  Ibs.  per  square  foot,  the  pressure 
in  pounds  resisting  the  motion  of  the  piston  at  any  point  in  the 
stroke  is 

62'4  .  A  .  h  Ibs. 

If  therefore,  VNST  be  measured  in  square  feet  the  work  done 
per  stroke  in  ft.-lbs. 

=  62-4  A.  VNST. 

261.    Head  lost  at  the  suction  valve. 

In  determining  the  pressure  head  H0  in  the  cylinder,  no  account 
has  been  taken  of  the  head  lost  due  to  the  sudden  enlargement 
from  the  pipe  into  the  cylinder,  or  of  the  more  serious  loss  of  head 
due  to  the  water  passing  through  the  valve.  It  is  probable  that  the 

whole  of  the  velocity  head,  ~  —  5  >  °^  ^ne  water  entering  the  cylinder 

from  the  pipe  is  lost  at  the  valve,  in  which  case  the  available  head 
H  will  not  only  have  to  give  this  velocity  to  the  water,  but  will 


RECIPROCATING   PUMPS  469 

v2 
also  have  to  give  a  velocity  head  ~-  to  any  water  entering  the 

cylinder  from  the  pipe. 

The  pressure  head  H0  in  the  cylinder  then  becomes 

v2   A"     tf     4/L^A2     F/       IA 
-h---~ 


262.  Variation  of  the  pressure  in  hydraulic  motors  due 
to  inertia  forces. 

The  description  of  hydraulic  motors  is  reserved  for  the  next 
chapter,  but  as  these  motors  are  similar  to  reversed  reciprocating 
pumps,  it  is  convenient  here  to  refer  to  the  effect  of  the  inertia 
forces  in  varying  the  effective  pressure  on  the  motor  piston. 

If  L  is  the  length  of  the  supply  pipe  of  a  hydraulic  motor,  a 
the  cross-sectional  area  of  the  supply,  A  the  cross-sectional  area 
of  the  piston  of  the  motor,  and  F  the  acceleration,  the  acceleration 

F.A 

of  the  water  in  the  pipe  is  —   —  and  the  head  required  to  accelerate 

a 

the  water  in  the  pipe  is 


ga 

If  p  is  the  pressure  per  square  foot  at  the  inlet  end  of  the 
supply  pipe,  and  hf  is  equal  to  the  losses  of  head  by  friction  in  the 
pipe,  and  at  the  valve  etc.,  when  the  velocity  of  the  piston  is  v,  the 
pressure  on  the  piston  per  square  foot  is 


When  the  velocity  of  the  piston  is  diminishing,  F  is  negative, 
and  the  inertia  of  the  water  in  the  pipe  increases  the  pressure  on 
the  piston. 

Example  (1).  The  stroke  of  a  double  acting  pump  is  15  inches  and  the  number  of 
strokes  per  minute  is  80.  The  diameter  of  the  plunger  is  12  inches  and  it  moves 
with  simple  harmonic  motion.  The  centre  of  the  pump  is  13  feet  above  the  water 
in  the  well  and  the  length  of  the  suction  pipe  is  25  feet. 

To  find  the  diameter  of  the  suction  pipe  that  no  separation  shall  take  place, 
assuming  it  to  take  place  when  the  pressure  head  becomes  less  than  7  feet. 

As  the  plunger  moves  with  simple  harmonic  motion,  it  may  be  supposed  driven 
by  a  crank  of  7£  inches  radius  and  a  very  long  connecting  rod,  the  angular 
velocity  of  the  crank  being  2?r40  radians  per  minute. 

The  acceleration  at  the  end  of  the  stroke  is  then 

47r2  .  402  .  r 


Therefore, 

from  which  —  =  1'64. 

a 


470  HYDRAULICS 

Therefore  d"  =  1  28 

and  d  =  9-4". 

Ar  is  clearly  less  than  al,  therefore  separation  cannot  take  place  at  any  other 
point  in  the  stroke. 

Example  (2).  The  pump  of  example  (1)  delivers  water  into  a  rising  main 
1225  feet  long  and  5  inches  diameter,  which  is  fitted  with  an  air  vessel. 

The  water  is  lifted  through  a  total  height  of  220  feet. 

Neglecting  all  losses  except  friction  in  the  delivery  pipe,  determine  the  horse- 
power required  to  work  the  pump.  /=-Ol05. 

Since  there  is  an  air  vessel  in  the  delivery  pipe  the  velocity  of  flow  u  will  be 
practically  uniform. 

Let  A  and  a  be  the  cross-sectional  areas  of  the  pump  cylinder  and  pipe  respect- 

A  .  2r  .  80     D2  2r  .  80 
Then>  —  60^=^~60- 

122    10    80 
=  25-¥-60 
The  head  h  lost  due  to  friction  is 

•042  x  9'62  x  1225 

*p-A 

=  176-4  feet. 
The  total  lift  is  therefore 

220  +  176-4  =  396-4  feet. 
The  weight  of  water  lifted  per  minute  is 

.         .  80  x  62-5  Ibs.  =  4900  Ibs. 


Example  (3).  If  in  example  (2)  the  air  vessel  is  near  the  pump  and  the  mean 
level  of  the  water  in  the  vessel  is  to  be  kept  at  2  feet  above  the  centre  of  the 
pump,  find  the  pressure  per  sq.  inch  in  the  air  vessel. 

The  head  at  the  junction  of  the  air  vessel  and  the  supply  pipe  is  the  head 
necessary  to  lift  the  water  207  feet  and  overcome  the  friction  of  the  pipe. 
Therefore,  Hw  +  2'  =  207  +  176-4, 

Hw=  381-4  feet, 

381-4  x  62-5 
**:  -144- 
=  165  Ibs.  per  sq.  inch. 

Example  (4).  A  single  acting  hydraulic  motor  making  50  strokes  per  minute 
has  a  cylinder  8  inches  diameter  and  the  length  of  the  stroke  is  12  inches.  The 
diameter  of  the  supply  pipe  is  3  inches  and  it  is  500  feet  long.  The  motor  is 
supplied  with  water  from  an  accumulator,  see  Fig.  339,  at  a  constant  pressure  of 
300  Ibs.  per  sq.  inch. 

Neglecting  the  mass  of  water  in  the  cylinder,  and  assuming  the  piston  moves 
with  simple  harmonic  motion,  find  the  pressure  on  the  piston  at  the  beginning  and 
the  centre  of  its  stroke.  The  student  should  draw  a  diagram  of  pressure  for  one 
stroke. 

There  are  25  useful  strokes  per  minute  and  the  volume  of  water  supplied 
per  minute  is,  therefore, 

25  .  ~  d2  =  8-725  cubic  feet. 
4 

K/VJ 

At  the  commencement  of  the  stroke  the  acceleration  is  ?r2  ^-  r,  and  the  velocity 
in  the  supply  pipe  is  zero. 


RECIPROCATING   PUMPS  471 

The  head  required  to  accelerate  the  water  in  the  pipe  is,  therefore, 

=  7T2 . 502 .  1 .  82 .  500 
a~      60a.2.32.32 

=  380  feet, 
which  is  equivalent  to  165  Ibs.  per  sq.  inch. 

The  effective  pressure  on  the  piston  is  therefore  135  Ibs.  per  sq.  inch. 
At  the   end   of  the   stroke  the   effective  pressure  on   the  piston   is  465  Ibs. 
per  sq.  inch. 

At  the  middle  of  the  stroke  the  acceleration  is  zero  and  the  velocity  of  the 
piston  is 

•|$  TIT  =1*31  feet  per  second. 
The  friction  head  is  then 

•04.1-312.82.500' 

2i7BTT~ 

=  15-2  feet. 
The  pressure  on  the  plunger  at  the  middle  of  the  stroke  is 

300  Ibs.  -  15'2*62'5=  293-4  Ibs.  per  sq.  inch. 

J.44 

The  mean  friction  head  during  the  stroke  is  f .  15-2  =  10*1  feet,  and  the  mean 
loss  of  pressure  is  4-4  Ibs.  per  sq.  inch. 

The  work  lost  by  friction  in  the  supply  pipe  per  stroke  is  4 '4 .  -  .  82 .  1K 

=  222  ft.  Ibs. 

The  work  lost  per  minute  =  5500  ft.  Ibs. 
The  net  work  done  per  minute  neglecting  other  losses  is 

(300  Ibs.  -4-4).^.Z8.82.25 

=  370,317  ft.  Ibs., 

and  therefore  the  work  lost  by  friction  is  comparatively  small,  being  less  than 
2  per  cent. 

Other  causes  of  loss  in  this  case,  are  the  loss  of  head  due  to  shock  where  the 
water  enters  the  cylinder,  and  losses  due  to  bends  and  contraction  at  the  valves. 

It  can  safely  be  asserted,  that  at  any  instant  a  head  equal  to  the  velocity  head, 
of  the  water  in  the  pipe,  will  be  lost  by  shock  at  the  valves,  and  a  similar  quantity 
at  the  entrance  to  the  cylinder.  These  quantities  are  however  always  small,  and 
even  if  there  are  bends  along  the  pipe,  which  cause  a  further  loss  of  head  equal  to 
the  velocity  head,  or  even  some  multiple  of  it,  the  percentage  loss  of  head  will  still 
be  small,  and  the  total  hydraulic  efficiency  will  be  high. 

This  example  shows  clearly  that  power  can  be  transmitted  hydraulically  very 
efficiently  over  comparatively  long  distances. 

263.     High  pressure  plunger  pump. 

Fig.  323  shows  a  section  through  a  high  pressure  pump 
suitable  for  pressures  of  700  or  800  Ibs.  per  sq.  inch. 

Suction  takes  place  on  the  outward  stroke  of  the  plunger,  and 
delivery  on  both  strokes. 

A  brass  liner  is  fitted  in  the  cylinder  and  the  plunger  which, 
as  shown,  is  larger  in  diameter  at  the  right  end  than  at  the  left, 
is  also  made  of  brass ;  the  piston  rod  is  of  steel.  Hemp  packing 
is  used  to  prevent  leakage  past  the  piston  and  also  in  the  gland 
box. 

The  plunger  may  have  leather  packing  as  in  Fig.  324. 

On  the  outward  stroke  neglecting  slip  the  volume  of  water 


472 


HYDRAULICS 


ffTl  fTll 


RECIPROCATING    PUMPS 


473 


drawn  into  the  cylinder  is  7  D02 .  L  cubic  feet,  D0  being  the  dia- 
meter of  the  piston  and  L  the  length  of  the  stroke.  The  quantity 
of  water  forced  into  the  delivery  pipe  through  the  valve  VD  is 

—  (D02  —  d2)  L  cubic  feet, 

\         4< 

d  being  the  diameter  of  the  small  part  of  the 
plunger. 

On  the  in-stroke,  the  suction  valve  is 
closed  and  water  is  forced  through  the 
delivery  valve;  part  of  this  water  enters 
the  delivery  pipe  and  part  flows  behind  the 
piston  through  the  port  P.  Yi%'  324- 

The  amount  that  flows  into  the  delivery  pipe  is 


If,  therefore,  (D02  -  d2)  is  made  equal  to  d2,  or  D0  is  \/2d,  the 
delivery,  during  each  stroke,  is  |  D02L  cubic  feet,  and  if  there  are 
n  strokes  per  minute,  the  delivery  is  42*45  D02Im  gallons  per 


minute. 


Fig.  325.     Tangye  Duplex  Pump. 

264.     Duplex  feed  pump. 

Fig.  325  shows  a  section  through  one  pump  and  steam  cylinder 
of  a  Tangye  double-acting  pump. 


474 


HYDRAULICS 


There  are  two  steam  cylinders  side  by  side,  one  of  which  only 
is  shown,  and  two  pump  cylinders  in  line  with  the  steam  cylinders. 

In  the  pump  the  two  lower  valves  are  suction  valves  and  the 
two  upper  delivery  valves.  As  the  pump  piston  P  moves  to  the 
right,  the  left-hand  lower  valve  opens  and  water  is  drawn  into  the 
pump  from  the  suction  chamber  C.  During  this  stroke  the  right 
upper  valve  is  open,  and  water  is  delivered  into  the  delivery  Ci. 
When  the  piston  moves  to  the  left,  the  water  is  drawn  in  through 
the  lower  right  valve  and  delivered  through  the  upper  left  valve. 

The  steam  engine  has  double  ports  at  each  end.  As  the  piston 
approaches  the  end  of  its  stroke  the  steam  valve,  Fig.  326,  is  at  rest 
and  covers  the  steam  port  1  while  the  "inner  steam  port  2  is  open 
to  exhaust.  When  the  piston  passes  the  steam  port  2,  the  steam 
enclosed  in  the  cylinder  acts  as  a  cushion  and  brings  the  piston 
and  plunger  gradually  to  rest. 


Fig.  326. 


Fig.  327. 


Let  the  one  engine  and  pump  shown  in  section  be  called  A  and 
the  other  engine  and  pump,  not  shown,  be  called  B. 

As  the  piston  of  A  moves  from  right  to  left,  the  lever  L,  Figs. 
325  and  327,  rotates  a  spindle  to  the  other  end  of  which  is  fixed  a 
crank  M,  which  moves  the  valve  of  the  cylinder  B  from  left  to 
right  and  opens  the  left  port  of  the  cylinder  B.  Just  before  the 
piston  of  A  reaches  the  left  end  of  its  stroke,  the  piston  of  B, 
therefore,  commences  its  stroke  from  left  to  right,  and  by  a  lever 
L!  and  crank  MI  moves  the  valve  of  cylinder  A  also  from  left  to 
right,  and  the  piston  of  A  can  then  commence  its  return  stroke. 
.t  should  be  noted  that  while  the  piston  of  A  is  moving,  that  of 
B  is  practically  at  rest,  and  vice  versa. 

265.     The  hydraulic  ram. 

The  hydraulic  ram  is  a  machine  which  utilises  the  momentum 
of  a  stream  of  water  falling  a  small  height  to  raise  a  part  of  the 
water  to  a  greater  height. 

In  the  arrangement  shown  in  Fig.  328  water  is  supplied  from  a 
tank,  or  stream,  through  a  pipe  A  into  a  chamber  B,  which  has  two 


PUMPS 


475 


valves  Y  and  Yi .  When  no  flow  is  taking  place  the  valve  V  falls 
off  its  seating  and  the  valve  Vi  rests  on  its  seating.  If  water  is 
allowed  to  flow  along  the  pipe  B  it  will  escape  through  the  open 
valve  Y.  The  contraction  of  the  jet  through  the  valve  opening, 
exactly  as  in  the  case  of  the  plate  obstructing  the  flow  in  a  pipe, 
page  168,  causes  the  pressure  to  be  greater  on  the  under  face  of 
the  valve,  and  when  the  pressure  is  sufficiently  large  the  valve 
will  commence  to  close.  As  it  closes  the  pressure  will  increase 
and  the  rate  of  closing  will  be  continually  accelerated.  The  rapid 
closing  of  the  valve  arrests  the  motion  of  the  water  in  the  pipe, 
and  there  is  a  sudden  rise  in  pressure  in  B,  which  causes  the 
valve  YI  to  open,  and  a  portion  of  the  water  passes  into  the  air 
vessel  C.  The  water  in  the  supply  pipe  and  in  the  vessel  B,  after 
being  brought  to  rest,  recoils,  like  a  ball  thrown  against  a  wall, 
and  the  pressure  in  the  vessel  is  again  diminished,  allowing  the 
water  to  once  more  escape  through  the  valve  Y.  The  cycle  of 
operations  is  then  repeated,  more,  water  being  forced  into  the  air 
chamber  C,  in  which  the  air  is  compressed,  and  water  is  forced  up 
the  delivery  pipe  to  any  desired  height. 


Fig.  328. 

Let  h  be  the  height  the  water  falls  to  the  ram,  H  the  height  to 
which  the  water  is  lifted. 

If  W  Ibs.  of  water  descend  the  pipe  per  second,  the  work 
available  per  second  is  Wh  foot  Ibs.,  and  if  e  is  the  efficiency  of  the 
ram,  the  weight  of  water  lifted  through  a  height  H  will  be 

W.h 


w  = 


e.H 


The  efficiency  e  diminishes  as  H  increases  and  may  be  taken  as 
60  per  cent,  at  high  heads. 

Fig.  329  shows  a  section  through  the  De  Cours  hydraulic 
ram,  the  valves  of  which  are  controlled  by  springs.  The  springs 


476 


HYDRAULICS 


can  be  regulated  so  that  the  number  of  beats  per  minute  is  com- 
pletely under  control,  and  can  be  readily  adjusted  to  suit  varying 

heads. 

With  this  type  of  ram  Messrs  Bailey  claim  to  have  obtained  at 
low  heads,  an  efficiency  of  more  than  90  per  cent.,  and  with  H 
equal  to  8h  an  efficiency  of  80  per  cent. 


Fig.  329.     De  Cours  Hydraulic  Earn. 

As  the  water  escapes  through  the  valve  Vi  into  the  air  vessel  C, 
a  little  air  should  be  taken  with  it  to  maintain  the  air  pressure  in 
C  constant. 

This  is  effected  in  the  De  Cours  ram  by  allowing  the  end  of  the 
exhaust  pipe  F  to  be  under  water.  At  each  closing  of  the  valve 


PUMPS 


477 


V,  the  siphon  action  of  the  water  escaping  from  the  discharge 
causes  air  to  be  drawn  in  past  the  spindle  of  the  valve.  A  cushion 
of  air  is  thus  formed  in  the  box  B  every  stroke,  and  some  of  this 
air  is  carried  into  C  when  the  valve  Vi  opens. 

The  extreme  simplicity  of  the  hydraulic  ram,  together  with 
the  ease  with  which*  it  can  be  adjusted  to  work  with  varying 
quantities  of  water,  render  it  particularly  suitable  for  pumping 
in  out-of-the-way  places,  and  for  supplying  water,  for  fountains 
and  domestic  purposes,  to  country  houses  situated  near  a  stream. 

266.     Lifting  water  by  compressed  air. 

A  very  simple  method  of  raising  water  from  deep  wells  is  by 
means  of  compressed  air.  A  delivery  pipe  is  sunk  into  a  well, 
the  open  end  of  the  pipe  being  placed  at  a  considerable  distance 
below  the  surface  of  the  water  in  the  well. 


AirTuube 


Fig.  330. 


Fig.  331. 


In  the  arrangement  shown  in  Fig.  330,  there  is  surrounding  the 
delivery  tube  a  pipe  of  larger  diameter  into  which  air  is  pumped 
by  a  compressor. 

The  air  rises  up  the  delivery  pipe  carrying  with  it  a  quantity  of 
water.  An  alternative  arrangement  is  shown  in  Fig.  331. 

Whether  the  air  acts  as  a  piston  and  pushes  the  water  in  front 
of  it,  or  forms  a  mixture  with  the  water,  according  to  Kelly*, 
depends  very  largely  upon  the  rate  at  which  air  is  supplied  to  the 
pump. 

In  the  pump  experimented  upon  by  Kelly,  at  certain  rates  of 

*  Proc.  List.  C.  E.  Vol.  CLXIII. 


478  HYDRAULICS 

working  the  discharge  was  continuous,  the  air  and  the  water  being 
mixed  together,  while  at  low  discharges  the  action  was  intermittent 
and  the  pump  worked  in  a  definite  cycle;  the  discharge  commenced 
slowly;  the  velocity  then  gradually  increased  until  the  pipe 
discharged  full  bore;  this  was  followed  by  a  rush  of  air,  after 
which  the  flow  gradually  diminished  and  finally  stopped ;  after  a 
period  of  no  flow  the  cycle  commenced  again.  When  the  rate  at 
which  air  was  supplied  was  further  diminished,  the  water  rose 
up  the  delivery  tube,  but  not  sufficiently  high  to  overflow,  and  the 
air  escaped  without  doing  useful  work. 

The  efficiency  of  these  pumps  is  very  low  and  only  in  exceptional 
cases  does  it  reach  50  per  cent.  The  volume  v  of  air,  in  cubic  feet, 
at  atmospheric  pressure,  required  to  lift  one  cubic  foot  of  water 
through  a  height  h  depends  upon  the  efficiency.  With  an  ef- 
ficiency of  30  per  cent,  it  is  approximately  v  =  ^)  an(i  with  an 

efficiency  of  40  per  cent,  v  =  ^  approximately. 

It  is  necessary  that  the  lower  end  of  the  delivery  be  at  a  greater 
distance  below  the  surface  of  the  water  in  the  well,  than  the  height 
of  the  lift  above  the  free  surface,  and  the  well  has  consequently  to 
be  made  very  deep. 

On  the  other  hand  the  well  is  much  smaller  in  diameter  than 
would  be  required  for  reciprocating  or  centrifugal  pumps,  and  the 
initial  cost  of  constructing  the  well  per  foot  length  is  considerably 
less. 

EXAMPLES. 

(1)  Find  the  horse-power  required  to  raise  100  cubic  feet  of  water  per 
minute  to  a  height  of  125  feet,  by  a  pump  whose  efficiency  is  ^. 

(2)  A  centrifugal  pump  has  an  inner  radius  of  4  inches  and  an  outer 
radius  of  12  inches.     The  angle  the  blade  makes  with  the  direction  of 
motion  at  exit  is  153  degrees.   The  wheel  makes  545  revolutions  per  minute. 

The  discharge  of  the  pump  is  3  cubic  feet  per  second.  The  sides  of  the 
wheel  are  parallel  and  2  inches  apart. 

Determine  the  inclination  of  the  tip  of  the  blades  at  inlet  so  that  there 
shall  be  no  shock,  the  velocity  with  which  the  water  leaves  the  wheel,  and 
the  theoretical  lift.  If  the  head  due  to  the  velocity  with  which  the  water 
leaves  the  wheel  is  lost,  find  the  theoretical  lift. 

(3)  A  centrifugal  pump  wheel  has  a  diameter  of  7  inches  and  makes 
1358  revolutions  per  minute. 

The  blades  are  formed  so  that  the  water  enters  and  leaves  the  wheel 
without  shock  and  the  blades  are  radial  at  exit.  The  water  is  lifted  by  the 
pump  29-4  feet.  Find  the  manometric  efficiency  of  the  pump. 


PUMPS  479 

(4)  A  centrifugal  pump  wheel  11  inches  diameter  which  runs  at  1203 
revolutions  per  minute  is   surrounded  by  a  vortex  chamber  22  inches 
diameter,  and  has  radial  blades  at  exit.     The  pressure  head  at  the  circum- 
ference of  the  wheel  is  23  feet.     The  water  is  lifted  to  a  height  of  43'5 
feet  above  the  centre  of  the  pump.     Find  the  efficiency  of  the  whirlpool 
chamber. 

(5)  The  radial  velocity  of  flow  through  a  pump  is  5  feet  per  second,  and 
the  velocity  of  the  outer  periphery  is  60  feet  per  second. 

The  angle  the  tangent  to  the  blade  at  outlet  makes  with  the  direction 
of  motion  is  120  degrees.  Determine  the  pressure  head  and  velocity  head 
where  the  water  leaves  the  wheel,  assuming  the  pressure  head  in  the  eye 
of  the  wheel  is  atmospheric,  and  thus  determine  the  theoretical  lift. 

(6)  A  centrifugal  pump  with  vanes  curved  back  has  an  outer  radius  of 
10  inches  and  an  inlet  radius  of  4  inches,  the  tangents  to  the  vanes  at  outlet 
being  inclined  at  40°  to  the  tangent  at  the  outer  periphery.     The  section  of 
the  wheel  is  such  that  the  radial  velocity  of  flow  is  constant,  5  feet  per 
second ;  and  it  runs  at  700  revolutions  per  minute. 

Determine : — 

(1)  the  angle  of  the  vane  at  inlet  so  that  there  shall  be  no  shock, 

(2)  the  theoretical  lift  of  the  pump, 

(3)  the  velocity  head  of  the  water  as  it  leaves  the  wheel.     Lond. 
Un.  1906. 

(7)  A  centrifugal  pump  4  feet  diameter  running  at  200  revolutions  per 
minute,  pumps  5000  tons  of  water  from  a  dock  in  45  minutes,  the  mean 
lift  being  20  feet.     The  area  through  the  wheel  periphery  is  1200  square 
inches  and  the  angle  of  the  vanes  at  outlet  is  26°.    Determine  the  hydraulic 
efficiency  and  estimate  the  average  horse -power.     Find  also  the  lowest 
speed  to  start  pumping  against  the  head  of  20  feet,  the  inner  radius  being 
half  the  outer.     Lond.  Un.  1906. 

(8)  A  centrifugal  pump,  delivery  1500  gallons  per  minute  with  a  lift  of 
25  feet,  has  an  outer  diameter  of  16  inches,  and  the  vane  angle  is  30°.     All 
the  kinetic  energy  at  discharge  is  lost,  and  is  equivalent  to  50  per  cent,  of 
the  actual  lift.     Find  the  revolutions  per  minute  and  the  breadth  at  the 
inlet,  the  velocity  of  whirl  being  half  the  velocity  of  the  wheel.     Lond. 
Un.  1906. 

(9)  A  centrifugal  pump  has  a  rotor  19|  inches  diameter ;  the  width  of 
the  outer  periphery  is  3T7^  inches.     Using  formula  (1),  section  236,  deter- 
mine the  discharge  of  the  pump  when  the  head  is  30  feet  and  Vi  is  50. 

(10)  The  angle  $  at  the  outlet  of  the  pump  of  question  (9)  is  13°. 
Find  the  velocity  with  which  the  water  leaves  the  wheel,  and  the 

minimum  proportion  of  the  velocity  head  that  must  be  converted  into  work, 
if  the  other  losses  are  15  per  cent,  and  the  total  efficiency  70  per  cent. 

(11)  The  inner  diameter  of  a  centrifugal  pump  is  12£  inches,  the  outer 
diameter  21|  inches.    The  width  of  the  wheel  at  outlet  is  3|  inches.    Using 
equation  (2),  section  236,  find  the  discharge  of  the  pump  when  the  head  is 
21'5  feet,  and  the  number  of  revolutions  per  minute  is  440. 


480  HYDRAULICS 

(12)  The  efficiency  of  a  centrifugal  pump  when  running  at  550  revolu- 
tions per  minute  is  70  per  cent.    The  mean  angle  the  tip  of  the  vane  makes 
with  the  direction  of  motion  of  the  inlet  edge  of  the  vane  is  99  degrees. 
The  angle  the  tip  of  the  vane  makes  with  the  direction  of  motion  of  the 
edge  of  the  vane  at  exit  is  167  degrees.     The  radial  velocity  of  flow  is  3'6 
feet  per  second.     The  internal  diameter  of  the  wheel  is  11|  inches  and  the 
external  diameter  19£  inches. 

Find  the  kinetic  energy  of  the  water  when  it  leaves  the  wheel. 

Assuming  that  5  per  cent,  of  the  energy  is  lost  by  friction,  and  that  one- 
half  of  the  kinetic  energy  at  exit  is  lost,  find  the  head  lost  at  inlet  when  the 
lift  is  30  feet.  Hence  find  the  probable  velocity  impressed  on  the  water  as 
it  enters  the  wheel. 

(13)  Describe  a  forced  vortex,  and  sketch  the  form  of  the  free  surface 
when  the  angular  velocity  is  constant. 

In  a  centrifugal  pump  revolving  horizontally  under  water,  the  diameter 
of  the  inside  of  the  paddles  is  1  foot,  and  of  the  outside  2  feet,  and  the 
pump  revolves  at  400  revolutions  per  minute.  Find  approximately  how 
high  the  water  would  be  lifted  above  the  tail  water  level. 

(14)  Explain  the  action  of  a  centrifugal  pump,  and  deduce  an  expression 
for  its  efficiency.     If  such  a  pump  were  required  to  deliver  1000  gallons  an 
hour  to  a  height  of  20  feet,  how  would  you  design  it  ?     Lond.  Un.  1903. 

(15)  Find  the  speed  of  rotation  of  a  wheel  of  a  centrifugal  pump  which 
is  required  to  lift  200  tons  of  water  5  feet  high  in  one  minute ;  having  given 
the  efficiency  is  0'6.     The  velocity  of  flow  through  the  wheel  is  4*5  feet  per 
second,  and  the  vanes  are  curved  backward  so  that  the  angle  between  their 
directions  and  a  tangent  to  the  circumference  is  20  degrees.     Lond.  Un. 
1905. 

(16)  A  centrifugal  pump  is  required  to  lift  2000  gallons  of  water  per 
minute  through  20  feet.     The  velocity  of  flow  through  the  wheel  is  7  feet 
per  second  and  the  efficiency  0'6.     The  angle  the  tip  of  the  vane  at  outlet 
makes  with  the  direction  of  motion  is  150  degrees.    The  outer  radius  of  the 
wheel  is  twice  the  inner.     Determine  the  dimensions  of  the  wheel. 

(17)  A  double-acting  plunger  pump  has  a  piston  6  inches   diameter 
and  the  length  of  the  strokes  is  12  inches.     The  gross  head  is  500  feet, 
and  the  pump  makes  80  strokas  per  minute.     Assuming  no  slip,  find  the 
discharge  and  horse-power  of  the  pump.    Find  also  the  necessary  diameter 
for  the  steam  cylinder  of  an  engine  driving  the  pump  direct,  assuming  the 
steam  pressure  is  100  Ibs.  per  square  inch,  and  the  mechanical  efficiency 
of  the  combination  is  85  per  cent. 

(18)  A  plunger  pump  is  placed  above  a  tank  containing  water  at  a 
temperature  of  200°  F.     The  weight  of  the  suction  valve  is  2  Ibs.  and  its 
diameter  l£  inches.     Find  the  maximum  height  above  the  tank  at  which 
the  pump  may  be  placed  so  that  it  will  draw  water,  the  barometer  standing 
at  30  inches  and  the  pump  being  assumed  perfect  and  without  clearance. 
(The  vapour  tension  of  water  at  200°  F.  is  about  11-6  Ibs.  per  sq.  inch.) 

(19)  A  pump  cylinder  is  8  inches  diameter  and  the  stroke  of  the  plunger 
is  one  foot.     Calculate  the  maximum  velocity,  and  the  acceleration  of  the 


PUMPS  481 

water  in  the  suction  and  delivery  pipes,  assuming  their  respective  diameters 
to  be  7  inches  and  5  inches,  the  motion  of  the  piston  to  be  simple  harmonic, 
and  the  piston  to  make  36  strokes  per  minute. 

(20)  Taking  the  data  of  question  (19)  calculate  the  work  done  on  the 
suction  stroke  of  the  pump, 

(1)  neglecting  the  friction  in  the  suction  pipe, 

(2)  including  the  friction  in  the  suction  pipe  and  assuming  that  the 

suction  pipe  is  25  feet  long  and  that/=0'01. 

The  height  of  the  centre  of  the  pump  above  the  water  in  the  sump  is 
18  feet. 

(21)  If  the  pump  in  question  (20)  delivers  into  a  rising  main  against 
a   head  of   120  feet,   and  if  the  length  of    the  main  itself  is   250  feet, 
find  the  total  work  done  per  revolution.     Assuming  the  pump  to  be  double 
acting,  find  the  i.  H.  p.  required  to  drive  the  pump,  the  efficiency  being  *72 
and  no  slip  in  the  pump.     Find  the  delivery  of  the  pump,  assuming  a  slip 
of  5  per  cent. 

(22)  The  piston  of  a  pump  moves  with  simple  harmonic  motion,  and  it 
is  driven  at  40  strokes  per  minute.     The  stroke  is  one  foot.     The  suction 
pipe  is  25  feet  long,  and  the  suction  valve  is  19  feet  above  the  surface  of  the 
water  in  the  sump.     Find  the  ratio  between  the  diameter  of  the  suction 
pipe  and  the  pump  cylinder,  so  that  no  separation  may  take  place  at  the. 
dead  points.     Water  barometer  34  feet. 

(23)  Two  double-acting  pumps  deliver  water  into  a  main  without  an 
air  vessel.     Each  is  driven  by  an  engine  with  a  fly-wheel  heavy  enough  to 
keep  the  speed  of  rotation  uniform,  and  the  connecting  rods  are  very  long. 

Let  Q  be  the  mean  delivery  of  the  pumps  per  second,  QL  the  quantity  of 
water  in  the  main.  Find  the  pressure  due  to  acceleration  (a)  at  the  begin- 
ning of  a  stroke  when  one  pump  is  delivering  water,  (b)  at  the  beginning 
of  the  stroke  of  one  of  two  double-acting  pumps  driven  by  cranks  at  right 
angles  when  both  are  delivering.  When  is  the  acceleration  zero  ? 

(24)  A  double-acting  horizontal  pump  has  a  piston  6  inches  diameter 
(the  diameter  of  the  piston  rod  is  neglected)  and  the  stroke  is  one  foot. 
The  water  is  pumped  to  a  height  of  250  feet  along  a  delivery  pipe  450  feet 
long  and  4^  inches  diameter.     An  air  vessel  is  put  on  the  delivery  pipe 
10  feet  from  the  delivery  valve. 

Find  the  pressure  on  the  pump  piston  at  the  two  ends  of  the  stroke 
when  the  pump  is  making  40  strokes  per  minute,  assuming  the  piston 
moves  with  simple  harmonic  motion  and  compare  these  pressures  with  the 
pressures  when  there  is  no  air  vessel.  /='0075. 

(25)  A  single  acting  hydraulic  motor  makes  160  strokes  per  minute  and 
moves  with  simple  harmonic  motion. 

The  motor  is  supplied  with  water  from  an  accumulator  in  which  the 
pressure  is  maintained  at  200  Ibs.  per  square  inch. 

The  cylinder  is  8  inches  diameter  and  12  inches  stroke.  The  delivery 
pipe  is  200  feet  long,  and  the  coefficient,  which  includes  loss  at  bends,  etc. 
may  be  taken  as/=0'2. 

L.  H.  31 


482  HYDRAULICS 

Neglecting  the  mass  of  the  reciprocating  parts  and  of  the  variable 
quantity  of  water  in  the  cylinder,  draw  a  curve  of  effective  pressure  on  the 
piston. 

(26)  The  suction  pipe  of  a  plunger  pump  is  35  feet  long  and  4  inches 
diameter,  the  diameter  of  the  plunger  is  6  inches  and  the  stroke  1  foot. 

The  delivery  pipe  is  2|  inches  diameter,  90  feet  long,  and  the  head  at 
the  delivery  valve  is  40  feet.  There  is  no  air  vessel  on  the  pump.  The 
centre  of  the  pump  is  12  feet  6  inches  above  the  level  of  the  water  in  the 
sump. 

Assuming  the  plunger  moves  with  simple  harmonic  motion  and  makes 
50  strokes  per  minute,  draw  the  theoretical  diagram  for  the  pump. 

Neglect  the  effect  of  the  variable  quantity  of  water  in  the  cylinder  and 
the  loss  of  head  at  the  valves. 

(27)  Will  separation  take  place  anywhere  in  the  delivery  pipe  of  the 
pump,  the  data  of  which  is  given  in  question  (26),  if  the  pipe  first  runs 
horizontally  for  50  feet  and  then  vertically  for  40,  or  rises  40  feet  im- 
mediately from  the  pump   and  then  runs  horizontally  for  50  feet,  and 
separation  takes  place  when  the  pressure  head  falls  below  5  feet  ? 

(28)  A  pump  has  three  single-acting  plungers  29^  inches   diameter 
driven  by  cranks  at  120  degrees  with  each  other.     The  stroke  is  5  feet  and 
the  number  of  strokes  per  minute  40.    The  suction  is  16  feet  and  the  length 
of  the  suction  pipe  is  22  feet.     The  delivery  pipe  is  3  feet  diameter  and 
350  feet  long.     The  head  at  the  delivery  valve  is  214  feet. 

Find  (a)  the  minimum  diameter  of  the  suction  pipe  so  that  there  is  no 
separation,  assuming  no  air  vessel  and  that  separation  takes  place  when 
the  pressure  becomes  zero. 

(b)  The  horse-power  of  the  pump  when  there  is  an  air  vessel  on  the 
delivery  very  near  to  the  pump.  /=  '007. 

[The  student  should  draw  out  three  cosine  curves  differing  in  phase  by 
120  degrees.  Then  remembering  that  the  pump  is  single  acting,  the 
resultant  curve  of  accelerations  will  be  found  to  have  maximum  positive 

o)V  A 
and  also  negative  values  of   •  every   60  degrees.     The  maximum 

acceleration  head  is  then  ha=  +^—^  -  . 


For  no  separation,  therefore,  a  =       n*  —  —  -  . 

lo(/  (o4  —  ID)  _J 


(29)  The  piston  of  a  double-acting  pump  is  5  inches  in  diameter  and 
the  stroke  is  1  foot.  The  delivery  pipe  is  4  inches  diameter  and  400  feet 
long  and  it  is  fitted  with  an  air  vessel  8  feet  from  the  pump  cylinder.  The 
water  is  pumped  to  a  height  of  150  feet.  Assuming  that  the  motion  of  the 
piston  is  simple  harmonic,  find  the  pressure  per  square  inch  on  the  piston 
at  the  beginning  and  middle  of  its  stroke  and  the  horse-power  of  the  pump 
when  it  makes  80  strokes  per  minute.  Neglect  the  effect  of  the  variable 
quantity  of  water  in  the  cylinder.  Lond.  Un.  1906. 


PUMPS  483 

(30)  The  plunger  of  a  pump  moves  with  simple  harmonic  motion. 
Find  the  condition  that  separation  shall  not  take  place  on  the  suction 
stroke  and  show  why  the  speed  of  the  pump  may  be  increased  if  an  air 
vessel  is  put  in  the  suction  pipe.     Sketch  an  indicator  diagram  showing 
separation.     Explain  "  negative  slip."     Lond.  Un.  1906. 

(31)  In  a  single-acting  force  pump,  the  diameter  of  the  plunger  is 
4  inches,  stroke  6  inches,  length  of  suction  pipe  63  feet,  diameter  of  suction 
pipe  2 1  inches,  suction  head  0'07  ft.     When  going  at  10  revolutions  per 
minute,  it  is  found  that  the  average  loss  of  head  per  stroke  between  the 
suction  tank  and  plunger  cylinder  is  0'23  ft.     Assuming  that  the  frictional 
losses  vary  as  the  square  of  the  speed,  find  the  absolute  head  on  the  suction 
side  of  the  plunger  at  the  two  ends  and  at  the  middle  of  the  stroke,  the 
revolutions  being  50  per  minute,  and  the  barometric  head  34  feet.     Draw  a 
diagram  of    pressures  on    the   plunger — simple   harmonic  motion  being 
assumed.     Lond.  Un.  1906. 

(32)  A   single-acting  pump  without  an   air  vessel  has   a  stroke    of 
7£  inches.     The  diameter  of  the  plunger  is  4  inches  and  of  the  suction 
pipe  3£  inches.     The  length  of  the  suction  pipe  is  12  feet,  and  the  centre 
of  the  pump  is  9  feet  above  the  level  in  the  sump. 

Determine  the  number  of  single  strokes  per  second  at  which  theoreti- 
cally separation  will  take  place,  and  explain  why  separation  will  actually 
take  place  when  the  number  of  strokes  is  less  than  the  calculated  value. 

(33)  Explain  carefully  the  use  of  an  air  vessel  in  the  delivery  pipe  of  a 
pump.     The  pump  of  question  (32)  makes  100  single  strokes  per  minute, 
and  delivers  water  to  a  height  of  100  feet  above  the  water  in  the  well 
through  a  delivery  pipe  1000  feet  long  and  2  inches  diameter.     Large  air 
vessels  being  put  on  the  suction  and  delivery  pipes  near  to  the  pump. 

On  the  assumption  that  all  losses  of  head  other  than  by  friction  in 
the  delivery  pipe  are  neglected,  determine  the  horse-power  of  the  pump. 
There  is  no  slip. 

(34)  A  pump  plunger  has  an  acceleration  of  8  feet  per  second  per 
second  when  at  the  end  of  the  stroke,  and  the  sectional  area  of  the  plunger 
is  twice  the  sectional  area  of  the  delivery  pipe.     The  delivery  pipe  is  152 
feet  long.     It  runs  from  the  pump  horizontally  for  a  length  of  45  feet,  then 
vertically  for  40  feet,  then  rises  5  feet,  on  a  slope  of  1  vertical  to  3  hori- 
zontal, and  finally  runs  in  a  horizontal  direction. 

Find  whether  separation  will  take  place,  and  if  so  at  which  section 
of  the  pipe,  if  it  be  assumed  that  separation  takes  place  when  the  pressure 
head  in  the  pipe  becomes  7  feet. 

(35)  A  pump  of  the  duplex  kind,  Fig.  325,  in  which  the  steam  piston  is 
connected  directly  to  the  pump  piston,  works  against  a  head  of  h  feet  of 
water,  the  head  being  supplied  by  a  column  of  water  in  the  delivery  pipe. 
The  piston  area  is  AQ,  the  plunger  area  A,  the  delivery  pipe  area  a,  the 
length  of  the  delivery  pipe  I  and  the  constant  steam  pressure  on  the  piston 
Po  Ibs.  per  square  foot.     The  hydraulic  resistance  may  be  represented  by 

Fv2 

-«— ,  v  being  the  velocity  of  the  plunger  and  F  a  coefficient. 

31—2 


484  HYDRAULICS 

Show  that  when  the  plunger  has  moved  a  distance  x  from  the  beginning 
of  the  stroke 

_e-^.  Lend.  Un.  1906. 


(36)  A  pump  valve  of  brass  has  a  specific  gravity  of  8|  with  a  lift  of 
^  foot,  the  stroke  of  the  piston  being  4  feet,  the  head  of  water  40  feet  and 
the  ratio  of  the  full  valve  area  to  the  piston  area  one-fifth. 

If  the  valve  is  neither  assisted  nor  meets  with  any  resistance  to  closing, 
find  the  time  it  will  take  to  close  and  the  "slip"  due  to  this  gradual  closing. 

Time  to  close  is  given  by  formula,  S=^2.    f=L^  X  32-2.    Lond.  Un.  1906. 

' 


CHAPTER  XI. 


HYDRAULIC  MACHINES. 


267.    Joints  and  packings  used  in  hydraulic  work. 

The  high  pressures  used  in  hydraulic  machinery  make  it 
necessary  to  use  special  precautions  in  making  joints. 

Figs.  332  and  333  show  methods  of  connecting  two  lengths  of 
pipe.  The  arrangement  shown  in  Fig.  332  is  used  for  small 


Left  hcuricL 
thread 


Right 
thread 


Fig.  332. 


Fig.  333. 


Fig.  334. 


486 


HYDRAULICS 


wrought-iron  pipes,  no  packing  being  required.  In  Fig.  333  the 
packing  material  is  a  gutta-percha  ring.  Fig.  336  shows  an 
ordinary  socket  joint  for  a  cast-iron  hydraulic  main.  To  make 
the  joint,  a  few  cords  of  hemp  or  tarred  rope  are  driven  into 
the  socket.  Clay  is  then  put  round  the  outside  of  the  socket  and 
molten  lead  run  in  it.  The  lead  is  then  jammed  into  the  socket 
with  a  caulking  tool.  Fig.  335  shows  various  forms  of  packing 
leathers,  the  applications  of  which  will  be  seen  in  the  examples 
given  of  hydraulic  machines. 


Neck  leather 


Cup  leather 
Fig.  335. 


Fig.  336. 


Hemp  twine,  carefully  plaited,  and  dipped  in  hot  tallow, 
makes  a  good  packing,  when  used  in  suitably  designed  glands 
(see  Fig.  339)  and  is  also  very  suitable  for  pump  buckets, 
Fig.  323.  Metallic  packings  are  also  used  as  shown  in  Figs.  337 
and  338. 


Fig.  337. 


Fig.  338. 


268.    The  accumulator. 

The  accumulator  is  a  device  used  in  connection  with  hydraulic 
machinery  for  storing  energy. 

In  the  form  generally  adopted  in  practice  it  consists  of  a  long 
cylinder  C,  Fig.  339,  in  which  slides  a  ram  B,  and  into  which  water 
is  delivered  from  pumps.  At  the  top  of  the  ram  is  fixed  a  rigid 
cross  head  which  carries,  by  means  of  the  bolts,  a  large  cylinder 
which  can  be  filled  with  slag  or  other  heavy  material,  or  it  may 
be  loaded  with  cast-iron  weights  as  in  Fig.  340.  The  water  is 


HYDRAULIC   MACHINES 


487 


Fig.  339.     Hydraulic  Accumulator. 


488  HYDRAULICS 

admitted  to  the  cylinder  at  any  desired  pressures  through  a  pipe 
connected  to  the  cylinder  by  the  flange  shown  dotted,  and  the 
weight  is  so  adjusted  that  when  the  pressure  per  sq.  inch  in 
the  cylinder  is  a  given  amount  the  ram  rises. 

If  d  is  the  diameter  of  the  ram  in  inches,  p  the  pressure 
in  Ibs.  per  sq.  inch,  and  h  the  height  in  feet  through  which  the 
ram  can  be  lifted,  the  weight  of  the  ram  and  its  load  is 


and  the  energy  that  can  be  stored  in  the  accumulator  is 

E  =  p.7#.fe  foot  Ibs. 
4 

The  principal  object  of  the  accumulator  is  to  allow  hydraulic 
machines,  or  lifts,  which  are  being  supplied  with  hydraulic  power 
from  the  pumps,  to  work  for  a  short  time  at  a  much  greater  rate 
than  the  pumps  can  supply  energy.  If  the  pumps  are  connected 
directly  to  the  machines  the  rate  at  which  the  pumps  can  supply 
energy  must  be  equal  to  the  rate  at  which  the  machines  are 
working,  together  with  the  rate  at  which  energy  is  being  lost  by 
friction,  etc.,  and  the  pump  must  be  of  such  a  capacity  as  to  supply 
energy  at  the  greatest  rate  required  by  the  machines,  and  the 
frictional  resistances.  If  the  pump  supplies  water  to  an  accumu- 
lator, it  can  be  kept  working  at  a  steady  rate,  and  during  the  time 
when  the  demand  is  less  than  the  pump  supply,  energy  can  be 
stored  in  the  accumulator. 

In  addition  to  acting  as  a  storer  of  energy,  the  accumulator 
acts  as  a  pressure  regulator  and  as  an  automatic  arrangement  for 
starting  and  stopping  the  pumps. 

When  the  pumps  are  delivering  into  a  long  main,  the  demand 
upon  which  is  varying,  the  sudden  cutting  off  of  the  whole  or 
a  part  of  the  demand  may  cause  such  a  sudden  rise  in  the  pressure 
as  to  cause  breakage  of  the  pipe  line,  or  damage  to  the  pump. 
With  an  accumulator  on  the  pipe  line,  unless  the  ram  is 
descending  and  is  suddenly  brought  to  rest,  the  pressure  cannot 
rise  very  much  higher  than  the  pressure  p  which  will  lift  the  ram. 

To  start  and  stop  the  pump  automatically,  the  ram  as  it 
approaches  the  top  of  its  stroke  moves  a  lever  connected  to 
a  chain  which  is  led  to  a  throttle  valve  on  the  steam  pipe  of  the 
pumping  engine,  and  thus  shuts  off  steam.  On  the  ram  again 
falling  below  a  certain  level,  it  again  moves  the  lever  and  opens 
the  throttle  valve.  The  engine  is  set  in  motion,  pumping  re- 
commences, and  the  accumulator  rises. 


HYDRAULIC   MACHINES  489 

Example.  A  hydraulic  crane  working  at  a  pressure  of  700  Ibs.  per  sq.  inch  has 
to  lift  30  cwts.  at  a  rate  of  200  feet  per  minute  through  a  height  of  50  feet,  once 
every  1£  minutes.  The  efficiency  of  the  crane  is  70  per  cent,  and  an  accumulator 
is  provided. 

Find  the  volume  of  the  cylinder  of  the  crane,  the  minimum  horse-power  for  the 
pump,  and  the  minimum  capacity  of  the  accumulator. 

Let  A  be  the  sectional  area  of  the  ram  of  the  crane  cylinder  in  sq.  feet  and  L 
the  length  of  the  stroke  in  feet. 

Then,  p  .  144.  A  .  L  x  0'70  =  30  x  112  x  50', 

30  x  112  x  50 
AL  =  V  =  0-70x  144x700 

=  2-38  cubic  feet. 
The  rate  of  doing  work  in  the  lift  cylinder  is 


per  minute, 


and  the  work  done  in  lifting  50  feet  is  240,000  ft.  Ibs.  Since  this  has  to  be  done 
once  every  one  and  half  minutes,  the  work  the  pump  must  supply  in  one  and  half 
minutes  is  at  least  240,000  ft.  Ibs.  ,  and  the  minimum  horse-power  is 

240,000 

=  33,000  x  1-5  =  ' 
The  work  done  by  the  pump  while  the  crane  is  lifting  is 


I'D 

The  energy  stored  in  the  accumulator  must  be,  therefore,  at  least  200,000  ft.  Ibs. 
Therefore,  if  Va  is  its  minimum  capacity  in  cubic  feet, 

Va  x  700  x  144  =  200,000, 
or  Va  =  2  cubic  feet  nearly. 

269.     Differential  accumulator*. 

Tweddell's  differential  accumulator,  shown  in  Fig.  340,  has  a 
fixed  ram,  the  lower  part  of  which  is  made  slightly  larger  than 
the  upper  by  forcing  a  brass  liner  upon  it.  A  cylinder  loaded 
with  heavy  cast-iron  weights  slides  upon  the  ram,  water-tight 
joints  being  made  by  means  of  the  cup  leathers  shown.  Water 
is  pumped  into  the  cylinder  through  a  pipe,  and  a  passage  drilled 
axially  along  the  lower  part  of  the  ram. 

Let  p  be  the  pressure  in  Ibs.  per  sq.  inch,  d  and  di  the  dia- 
meters of  the  upper  and  lower  parts  of  the  ram  respectively. 
The  weight  lifted  (neglecting  friction)  is  then 


and  if  h  is  the  lift  in  feet,  the  energy  stored  is 
E  -  p.  ^W-d2)  h.  foot  Ibs. 

The  difference  of  the  diameters  d^  and  d  being  small,  the  pres- 
sure p  can  be  very  great  for  a  comparatively  small  weight  W. 

The  capacity  of  the  accumulator  is,  however,  very  small. 
This  is  of  advantage  when  being  used  in  connection  with 

*  Proceedings  Inst.  Mech.  Engs.,  1874. 


490 


HYDRAULICS 


Fig.  340. 


Fig.  341.     Hydraulic  Intensifies 


HYDRAULIC    MACHINES  491 

hydraulic  riveters,  as  when  a  demand  is  made  upon  the  ac- 
cumulator, the  ram  falls  quickly,  but  is  suddenly  arrested  when 
the  ram  of  the  riveter  comes  to  rest,  and  there  is  a  consequent 
increase  in  the  pressure  in  the  cylinder  of  the  riveter  which 
clinches  the  rivet.  Mr  Tweddell  estimates  that  when  the  ac- 
cumulator is  allowed  to  fall  suddenly  through  a  distance  of  from 
18  to  24  inches,  the  pressure  is  increased  by  50  per  cent. 

270.  Air  accumulator. 

The  air  accumulator  is  simply  a  vessel  partly  filled  with  air  and 
into  which  the  pumps,  which  are  supplying  power  to  machinery, 
deliver  water  while  the  machinery  is  not  at  work. 

Such  an  air  vessel  has  already  been  considered  in  connection 
with  reciprocating  pumps  and  an  application  is  shown  in  connection 
with  a  forging  press,  Fig.  343. 

If  V  is  the  volume  of  air  in  the  vessel  when  the  pressure  is 
p  pounds  per  sq.  inch  and  a  volume  v  of  water  is  pumped  into 
the  vessel,  the  volume  of  air  is  (V  — v). 

Assuming  the  temperature  remains  constant,  the  pressure  pi  in 
the  vessel  will  now  be 

p.V 

*"V^' 

If  V  is  the  volume  of  air,  and  a  volume  of  water  v  is  taken  out 
of  the  vessel, 

p.V 
P'  =  VTV 

271.  Intensifies. 

It  is  frequently  desirable  that  special  machines  shall  work  at 
a  higher  pressure  than  is  available  from  the  hydraulic  mains.  To 
increase  the  pressure  to  the  desired  amount  the  intensifier  is  used. 

One  form  is  shown  in  Fig.  341.  A  large  hollow  ram  works  in 
a  fixed  cylinder  C,  the  ram  being  made  water-tight  by  means  of  a 
stuffing-box.  Connected  to  the  cylinder  by  strong  bolts  is  a  cross 
head  which  has  a  smaller  hollow  ram  projecting  from  it,  and 
entering  the  larger  ram,  in  the  upper  part  of  which  is  made  a 
stuffing-box.  Water  from  the  mains  is  admitted  into  the  large 
cylinder  and  also  into  the  hollow  ram  through  the  pipe  and 
the  lower  valve  respectively  shown  in  Fig.  342. 

If  p  Ibs.  per  sq.  inch  is  the  pressure  in  the  main,  then  on 
the  underside  of  the  large  ram  there  is  a  total  force  acting 

of  p  7  D2  pounds,  and  the  pressure  inside  the  hollow  ram  rises  to 

D2 

p-r2  pounds  per  sq.  inch,  D  and  d  being  the  external  diameters 
cL 

of  the  large  ram  and  the  small  ram  respectively. 


492 


HYDRAULICS 


The  form  of  intensifier  here  shown  is  used  in  connection  with 
a  large  flanging  press.  The  cylinder  of  the  press  and  the  upper 
part  of  the  intensifier  are  filled  with  water  at  700  Ibs.  per  sq.  inch 
and  the  die  brought  to  the  work.  Water  at  the  same  pressure  is 
admitted  below  the  large  ram  of  the  intensifier  and  the  pressure 
in  the  upper  part  of  the  intensifier,  and  thus  in  the  press  cylinder, 
rises  to  2000  Ibs.  per  sq.  inch,  at  which  pressure  the  flanging 
is  finished. 


ToSjnaJL 

Cylinder 

of  Intensifier 


To  press. 
atWOOlbs. 
per  sq.  inch, 


Tb  Large  Cylinder  erf  Intensifier 


From  Pumps 


NOTL  Return  Valves  for 
Intensifier. 

Fig.  342. 


atlOOWs. 

persq.  inch. 


272.  Steam  intensifiers. 

The  large  cylinder  of  an  intensifier  may  be  supplied  with 
steam,  instead  of  water,  as  in  Fig.  343,  which  shows  a  steam  in- 
tensifier used  in  conjunction  with  a  hydraulic  forging  press.  These 
intensifiers  have  also  been  used  on  board  ship*  in  connection  with 
hydraulic  steering  gears. 

273.  Hydraulic  forging  press,  with  steam  intensifier  and 
air  accumulator. 

The  application  of  hydraulic  power  to  forging  presses  is  illus- 
trated in  Fig.  343.  This  press  is  worked  in  conjunction  with  a 
steam  intensifier  and  air  accumulator  to  allow  of  rapid  working. 
The  whole  is  controlled  by  a  single  lever  K,  and  the  press  is 
capable  of  making  80  working  strokes  per  minute. 

When  the  lever  K  is  in  the  mid  position  everything  is  at  rest ; 

on  moving  the  lever  partly  to  the  right,  steam  is  admitted  into  the 

cylinders  D  of  the  press  through  a  valve.     On  moving  the  lever  to 

its  extreme  position,  a  finger  moves  the  valve  M  and  admits  water 

*  Proceedings  Inst.  Mech.  Engs.,  1874. 


HYDRAULIC   MACHINES 


493 


under  a  relay  piston  shown  at  the  top  of  the  figure,  which  opens 
a  valve  E  at  the  top  of  the  air  vessel.  In  small  presses  the  valve 
E  is  opened  by  levers.  The  ram  B  now  ascends  at  the  rate  of 


about  1  foot  per  second,  the  water  in  the  cylinder  c  being  forced 
into  the  accumulator.  On  moving  the  lever  K  to  the  left,  as  soon 
as  it  has  passed  the  central  position  the  valve  L  is  opened  to 


494 


HYDRAULICS 


exhaust,  and  water  from  the  air  vessel,  assisted  by  gravity,  forces 
down  the  ram  B,  the  velocity  acquired  being  about  2  feet  per 
second,  until  the  press  head  A  touches  the  work.  The  movement 
of  the  lever  K  being  continued,  a  valve  situated  above  the  valve 
J  is  opened,  and  steam  is  admitted  to  the  intensifier  cylinder  H ; 
the  valve  E  closes  automatically,  and  a  large  pressure  is  exerted 
on  the  work  under  the  press  head. 

If  only  a  very  short  stroke  is  required,  the  bye-pass  valve  L  is 
temporarily  disconnected,  so  that  steam  is  supplied  continuously 
to  the  lifting  cylinders  D.  The  lever  K  is  then  simply  used  to 
admit  and  exhaust  steam  from  the  intensifier  H,  and  no  water 
enters  or  leaves  the  accumulator.  An  automatic  controlling  gear 
is  also  fitted,  which  opens  the  valve  J  sufficiently  early  to  prevent 
the  intensifier  from  overrunning  its  proper  stroke. 


Fig.  346. 


Fig.  344.  Fig.  345. 

274.     Hydraulic  cranes. 

Fig.  344  shows  a  section  through,  and  Fig.  345  an  elevation 
of,  a  hydraulic  crane  cylinder. 


HYDRAULIC   MACHINES  495 

One  end  of  a  wire  rope,  or  chain,  is  fixed  to  a  lug  L  on  the 
cylinder,  and  the  rope  is  then  passed  alternately  round  the  upper 
and  lower  pulleys,  and  finally  over  the  pulley  on  the  jib  of  the 
crane,  Fig.  346.  In  the  crane  shown  there  are  three  pulleys  on 
the  ram,  and  neglecting  friction,  the  pressure  on  the  ram  is  equally 
divided  among  the  six  ropes.  The  weight  lifted  is  therefore  one- 
sixth  of  the  pressure  on  the  ram,  but  the  weight  is  lifted  a  distance 
equal  to  six  times  the  movement  of  the  ram. 

Let  the  number  of  pulleys  on  the  end  of  the  ram  of  any  crane 

be  -a  ,  arranged  as  in  Fig.  347. 

The  movement  of  the  weight  will  then  be  n  times  that  of 
the  ram. 

Let  p  be  the  pressure  in  Ibs.  per  sq.  inch  in  the  cylinder  and 
d  the  diameter  of  the  ram  in  inches. 

The  pressure  on  the  ram  is 


and  the  energy  supplied  to  the  crane  per  foot  travel  of  the  ram  is 
therefore  P  foot  pounds. 

The  energy  supplied  per  unit  volume  displacement  is  144  .  p. 

The  actual  weight  lifted  is 

W  =  e  ^-  pd2  Ibs., 
4m, 

e  being  the  efficiency. 

When  full  load  is  being  lifted  e  is  between  0*7  and  0*8. 

For  a  given  lift  of  the  weight,  the  number  of  cubic  feet  of  water 
used,  and  consequently  the  energy  supplied,  is  the  same  whatever 
the  load  lifted,  and  at  light  loads  the  efficiency  is  very  small. 

275.     Double  power  cranes. 

To  enable  a  crane  designed  for  heavy  work  to  lift  light  loads 
with  reasonable  efficiency,  two  lifting  rams  of  different  diameters 
are  employed,  the  smaller  of  which  can  be  used  at  light  loads. 

A  convenient  arrangement  is  as  shown  in  Figs.  348  and  349, 
the  smaller  ram  R'  working  inside  the  large  ram  R. 

When  light  loads  are  to  be  lifted,  the  large  ram  is  prevented 
from  moving  by  strong  catches  C,  and  the  volume  of  water  used 
is  only  equal  to  the  diameter  of  the  small  ram  into  the  length  of 
the  stroke.  For  large  loads,  the  catches  are  released  and  the 
two  rams  move  together. 

Another  arrangement  is  shown  in  Fig.  350,  water  being  ad- 
mitted to  both  faces  of  the  piston  when  light  loads  are  to  be 
lifted,  and  to  the  face  A  only  when  heavy  loads  are  to  be  raised. 


496 


HYDRAULICS 


.SP 


HYDRAULIC   MACHINES 


497 


For  a  given  stroke  s  of  the  ram,  the  energy  supplied  in  the 
first  case  is 


;.  Ibs., 


and  in  the  second  case 


.  Ibs. 


Fig.  350.     Armstrong  Double-power  Hydraulic  Crane  Cylinder. 

276.     Hydraulic  crane  valves. 

Figs.  351  and  352  show  two  forms  of  lifting  and  lowering 
valves  used  by  Armstrong,  Whitworth  and  Co.  for  hydraulic 
cranes. 

In  the  arrangement  shown  in  Fig.  351  there  are  two  inde- 
pendent valves,  the  one  on  the  left  being  the  pressure,  and  that 
on  the  right  the  exhaust  valve. 


Fig.  351.     Armstrong-Whitworth 
Hydraulic  Crane  Valve. 

L.   H. 


Fig.  352.     Armstrong-Whitworth 
Hydraulic  Crane  Slide  Valve. 

32 


498  HYDRAULICS 

In  the  arrangement  shown  in  Fig.  352  a  single  D  slide  valve  s 
used.  Water  enters  the  valve  chest  through  the  pressure  passage 
P.  The  valve  is  shown  in  the  neutral  position.  If  the  valve 
is  lowered,  the  water  enters  the  cylinder,  but  if  it  is  right, 
water  escapes  from  the  cylinder  through  the  port  of  the  slide 
valve. 

277.  Small  hydraulic  press.  Fig.  353  is  a  section  through 
the  cylinder  of  a  small  hydraulic  press,  used  for  testing  springs. 

The  cast-iron  cylinder  is  fitted  with  a  brass  liner,  and  axially 
with  the  cylinder  a  rod  Pr,  having  a  piston  P  at  the  free  end, 
is  screwed  into  the  liner.  The  steel  ram  is  hollow,  the  inner 
cylinder  being  lined  with  a  brass  liner. 

Water  is  admitted  and  exhausted  from  the  large  cylinder 
through  a  Luthe  valve,  fixed  to  the  top  of  the  cylinder  and 
operated  by  the  lever  A.  The  small  cylinder  inside  the  ram  is 
connected  directly  to  the  pressure  pipe  by  a  hole  drilled  along  the 
rod  Pr,  so  that  the  full  pressure  of  the  water  is  continuously 
exerted  upon  the  small  piston  P  and  the  annular  ring  RR. 

Leakage  to  the  main  cylinder  is  prevented  by  means  of  a 
gutta-percha  ring  Gr  and  a  ring  leather  c,  and  leakage  past  the 
steel'  ram  and  piston  P  by  cup  leathers  L  and  LI  . 

When  the  valve  spindle  is  moved  to  the  right,  the  port  p  is 
connected  with  the  exhaust,  and  the  ram  is  forced  up  by  the 
pressure  of  the  water  on  the  annular  ring  RR.  On  moving  the 
valve  spindle  over  to  the  left,  pressure  water  is  admitted  into  the 
cylinder  and  the  ram  is  forced  down.  Immediately  the  pressure 
is  released,  the  ram  comes  back  again. 

Let  D  be  the  diameter  of  the  ram,  d  the  diameter  of  the 
rod  Pr,  di  the  diameter  of  the  piston  P,  and  p  the  water  pressure 
in  pounds  per  sq.  inch. 

The  resultant  force  acting  on  the  ram  is 


and  the  force  lifting  the  ram  when  pressure  is  released  from  the 
main  cylinder  is, 


The  cylindrical  valve  spindle  S,  has  a  chamber  C  cast  in  it, 
and  two  rings  of  six  holes  in  each  ring  are  drilled  through 
the  external  shell  of  the  chamber.  These  rings  of  holes  are  at 
such  a  distance  apart  that,  when  the  spindle  is  moved  to  the 
right,  one  ring  is  opposite  to  the  exhaust  and  the  other  opposite 
to  the  port  p,  and  when  the  spindle  is  moved  to  the  left,  the  holes 


HYDRAULIC   MACHINES 


499 


are  respectively  opposite  to  the  port  p  and  the  pressure  water 
inlet. 

Leakage  past  the  spindle  is  prevented  by  the  four  ring  leathers 
shown. 


Fig.  353.     Hydraulic  Press  with  Luthe  Valve. 

278.     Hydraulic  riveter. 

A  section  through  the  cylinder  and  ram  of  a  hydraulic  riveter 
is  shown  in  Fig.  354. 

32—2 


500 


HYDRAULICS 


Fig.  354.     Hydraulic  Riveter. 


Spring 
forclosuLq 
" 


Fig.  355.    Valves  for  Hydraulic  Eiveter. 


HYDRAULIC  MACHINES 


501 


The  mode  of  working  is  exactly  the  same  as  that  of  the  small 
press  described  in  section  277. 

An  enlarged  section  of  the  valves  is  shown  in  Fig.  355.  On 
pulling  the  lever  L  to  the  right,  the  inlet  valve  V  is  opened,  and 
pressure  water  is  admitted  to  the  large  cylinder,  forcing  out 
the  ram.  When  the  lever  is  in  mid  position,  both  valves  are 
closed  by  the  springs  S,  and  on  moving  the  lever  to  the  left,  the 
exhaust  valve  Vi  is  opened,  allowing  the  water  to  escape  from  the 
cylinder.  The  pressure  acting  on  the  annular  ring  inside  the 
large  ram  then  brings  back  the  ram.  The  methods  of  preventing 
leakage  are  clearly  shown  in  the  figures. 

279.     Hydraulic  engines. 

Hydraulic  power  is  admirably  adapted  for  machines  having  a 
reciprocating  motion  only,  especially  in  those  cases  where  the  load 
is  practically  constant. 


Fig.  356.     Hydraulic  Capstan. 


502 


HYDRAULICS 


It  has  moreover  been  successfully  applied  to  the  driving  of 
machines  such  as  capstans  and  winches  in  which  a  reciprocating 
motion  is  converted  into  a  rotary  motion. 

The  hydraulic-engine  shown  in  Figs.  356  and  357,  has  three 
cylinders  in  one  casting,  the  axes  of  which  meet  on  the  axis  of  the 
crank  shaft  S.  The  motion  of  the  piston  P  is  transmitted  to  the 
crank  pin  by  short  connecting  rods  R.  Water  is  admitted  and 
exhausted  through  a  valve  Y,  and  ports  p. 


Fig.  357. 


The  face  of  the  valve  is  as  shown  in  Fig.  358,  E  being  the 
exhaust  port  connected  through  the  centre  of  the  valve  to  the 
exhaust  pipe,  and  KM  the  pressure  port,  connected  to  the  supply 
chamber  H  by  a  small  port  through  the  side  of  the  valve.  The 
valve  seating  is  generally  made  of  lignum-vitae,  and  has  three 
circular  ports  as  shown  dotted  in  Fig.  358.  The  valve  receives  its 
motion  from  a  small  auxiliary  crank  T,  revolved  by  a  projection 
from  the  crank  pin  Gr.  When  the  piston  1  is  at  the  end  of  its 
stroke,  Fig.  359,  the  port  pi  should  be  just  opening  to  the  pressure 
port,  and  just  closing  to  the  exhaust  port  E.  The  port  ps  should 
be  fully  open  to  pressure  and  port  p2  fully  open  to  exhaust. 
When  the  crank  has  turned  through  60  degrees,  piston  3  will 


HYDRAULIC   MACHINES 


503 


be  at  the  inner  end  of  its  stroke,  and  the  edge  M  of  the  pressure 
port  should  be  just  closing  to  the  port  p3.  At  the  same  instant  the 
edge  N  of  the  exhaust  port  should  be  coincident  with  the  lower 
edge  of  the  port  ps.  The  angles  QOM,  and  LON,  therefore, 
should  each  be  60  degrees.  A  little  lead  may  be  given  to  the 
valve  ports,  i.e.  they  may  be  made  a  little  longer  than  shown  in 
the  Fig.  358,  so  as  to  ensure  full  pressure  on  the  piston  when 
commencing  its  stroke.  There  is  no  dead  centre,  as  in  whatever 
position  the  crank  stops  one  or  more  of  the  pistons  can  exert  a 
turning  moment  on  the  shaft,  and  the  engine  will,  therefore,  start 
in  any  position. 


Fig.  358. 


Fig.  359. 


The  crank*  effort,  or  turning  moment  diagram,  is  shown  in 
Fig.  359,  the  turning  moment  for  any  crank  position  OK  being 
OM.  The  turning  moment  can  never  be  less  than  ON,  which  is 
the  magnitude  of  the  moment  when  any  one  of  the  pistons  is  at 
the  end  of  its  stroke. 

This  type  of  hydraulic  engine  has  been  largely  used  for  the 
driving  of  hauling  capstans,  and  other  machinery  which  works 
intermittently.  It  has  the  disadvantage,  already  pointed  out  in 
connection  with  hydraulic  lifts  and  cranes,  that  the  amount  of 
water  supplied  is  independent  of  the  effective  work  done  by  the 
machine,  and  at  light  loads  it  is  consequently  very  inefficient. 
There  have  been  many  attempts  to  overcome  this  difficulty, 
notably  as  in  the  Hastie  engine  t,  and  Rigg  engine. 

*  See  text  book  on  Steam  Engine. 

t  Proceedings  Inst.  Mech.  Engs. ,  1874. 


504 


HYDRAULICS 


280.     Rigg  hydraulic  engine. 

To  adapt  the  quantity  of  water  used  to  the  work  done,  Rigg* 
has  modified  the  three  cylinder  engine  by  fixing  the  crank  pin,  and 
allowing  the  cylinders  to  revolve  about  it  as  centre. 

The  three  pistons  Pi,  P2  and  P3  are  connected  to  a  disc, 
Fig.  360,  by  three  pins.  This  disc  revolves  about  a  fixed  centre  A. 
The  three  cylinders  rotate  about  a  centre  Gr,  which  is  capable  of 
being  moved  nearer  or  further  away  from  the  point  A  as  desired. 
The  stroke  of  the  pistons  is  twice  AGr,  whether  the  crank  or  the 
cylinders  revolve,  and  since  the  cylinders,  for  each  stroke,  have  to 
be  filled  with  high  pressure  water,  the  quantity  of  water  supplied 
per  revolution  is  clearly  proportional  to  the  length  AGr. 


Fig.  360.     Kigg  Hydraulic  Engine. 

The  alteration  of  the  length  of  the  stroke  is  effected  by  means 
of  the  subsidiary  hydraulic  engine,  shown  in  Fig.  361.  There  are 
two  cylinders  C  and  Ci,  in  which  slide  a  hollow  double  ended 
ram  PPl  which  carries  the  pin  Gr,  Fig.  360.  Cast  in  one  piece  with 
the  ram  is  a  valve  box  B.  R  is  a  fixed  ram,  and  through  it  water 
enters  the  cylinder  Ci,  in  which  the  pressure  is  continuously 
maintained.  The  difference  between  the  effective  areas  of  P  and 
PI  when  water  is  in  the  two  cylinders,  is  clearly  equal  to  the  area 
of  the  ram  head  RL 

*  See  also  Engineer,  Vol.  LXXXV,  1898. 


HYDRAULIC   MACHINES 


505 


From  the  cylinder  Ci  the  water  is  led  along  the  passages 
shown  to  the  valve  V.  On  opening  this  valve  high-pressure 
water  is  admitted  to  the  cylinder  C.  A  second  valve  similar  to 
V,  but  not  shown,  is  used  to  regulate  the  exhaust  from  the 
cylinder  C.  When  this  valve  is  opened,  the  ram  PPj  moves  to 
the  left  and  carries  with  it  the  pin  Gr,  Fig.  360.  On  the  exhaust 
being  closed  and  the  valve  V  opened,  the  full  pressure  acts  upon 
both  ends  of  the  ram,  and  since  the  effective  area  of  P  is  greater 
than  PI  it  is  moved  to  the  right  carrying  the  pin  Gr.  If  both 
valves  are  closed,  water  cannot  escape  from  the  cylinder  C  and 
the  ram  is  locked  in  position  by  the  pressure  on  the  two  ends. 


To  Governor 


Water 
inlets 


Fig.  361. 


EXAMPLES. 

(1)  The  ram  of  a  hydraulic  crane  is  7  inches  diameter.     Water  is 
supplied  to  the  crane  at  700  Ibs.  per  square  inch.     By  suitable  gearing  the 
load  is  lifted  6  times  as  quickly  as  the  ram.     Assuming  the  total  efficiency 
of  the  crane  is  70  per  cent.,  find  the  weight  lifted. 

(2)  An  accumulator  has  a  stroke  of  23  feet ;  the  diameter  of  the  ram  is 
23  inches;  the  working  pressure  is  700  Ibs.  per  square  inch.     Find  the 
capacity  of  the  accumulator  in  horse-power  hours. 

(3)  The  total  weight  on  the  cage  of  an  ammunition  hoist  is  3250  Ibs. 
The  velocity  ratio  between  the  cage  and  the  ram  is  six,  and  the  extra  load 
on  the  cage  due  to  friction  may  be  taken  as  30  per  cent,  of  the  load  on  the 
cage.    The  steady  speed  of  the  ram  is  6  inches  per  second  and  the  available 
pressure  at  the  working  valve  is  700  Ibs.  per  square  inch. 

Estimate  the  loss  of  head  at  the  entrance  to  the  ram  cylinder,  and 
assuming  this  was  to  be  due  to  a  sudden  enlargement  in  passing  through 
the  port  to  the  cylinder,  estimate,  on  the  usual  assumption,  the  area  of  the 
port,  the  ram  cylinder  being  9|  inches  diameter.  Lond.  Un.  1906. 

3250x1-3x6 
The  effective  pressure     p  = • 


506  HYDRAULICS 

Loss  of  head 


w  2g 

v= velocity  through  the  valve. 


Area  of  port 

(4)  Describe,  with  sketches,  some  form  of  hydraulic  accumulator  suit- 
able for  use  in  connection  with  riveting.     Explain  by  the  aid  of  diagrams, 
if  possible,  the  general  nature  of  the  curve  of  pressure  on  the  riveter  ram 
during  the  stroke ;  and  point  out  the  reasons  of  the  variations.     Lond.  Un. 
1905.     (See  sections  262  and  269.) 

(5)  Describe  with  sketches  a  hydraulic  intensifier. 

An  intensifier  is  required  to  increase  the  pressure  of  700  Ibs.  per  square 
inch  on  the  mains  to  3000  Ibs.  per  square  inch.  The  stroke  of  the  intensi- 
fier is  to  be  4  feet  and  its  capacity  three  gallons.  Determine  the  diameters 
of  the  rams.  Inst.  C.  E.  1905. 

(6)  Sketch  in  good  proportion  a  section  through  a  differential  hydraulic 
accumulator.    What  load  would  be  necessary  to  produce  a  pressure  of  1  ton 
per  square  inch,  if  the  diameters  of  the  two  rams  are  4  inches  and  4^  inches 
respectively  ?    Neglect  the  friction  of  the  packing.    Give  an  instance  of  the 
use  of  such  a  machine  and  state  why  accumulators  are  used. 

(7)  A  Tweddell's  differential  accumulator  is  supplying  water  to  riveting 
machines.     The  diameters  of  the  two  rams  are  4  inches  and  4^  inches 
respectively,  and  the  pressure  in  the  accumulator  is  1  ton  per  square  inch. 
Suppose  when  the  valve  is  closed  the  accumulator  is  falling  at  a  velocity 
of  5  feet  per  second,  and  the  time  taken  to  bring  it  to  rest  is  2  seconds,  find 
the  increase  in  pressure  in  the  pipe. 

(8)  A  lift  weighing  12  tons  is  worked  by  water  pressure,  the  pressure 
in  the  main  at  the  accumulator  being  1200  Ibs.  per  square  inch ;  the  length 
of  the  supply  pipe  which  is  3|  inches  in  diameter  is  900  yards.     What  is 
the  approximate  speed  of  ascent  of  this  lift,  on  the  assumption  that  the 
friction  of  the  stuffing-box,  guides,  etc.  is  equal  to  6  per  cent,  of  the  gross 
load  lifted  and  the  ram  is  8  inches  diameter  ? 

(9)  Explain  what  is  meant  by  the  "  coefficient  of  hydraulic  resistance  " 
as  applied  to  a  whole  system,  and  what  assumption  is  usually  made  regard- 
ing it  ?     A  direct  acting  lift  having  a  ram  10  inches  diameter  is  supplied 
from  an  accumulator  working  under  a  pressure  of  750  Ibs.  per  square  inch. 
When  carrying  no  load  the  ram  moves  through  a  distance  of  60  feet,  at  a 
uniform  speed,  in  one  minute,  the  valves  being  fully  open.     Estimate  the 
coefficient  of  hydraulic  resistance  referred  to  the  velocity  of  the  ram,  and 
also  how  long  it  would  take  to  move  the  same  distance  when  the  ram 
carries  a  load  of  20  tons.     Lond.  Un.  1905. 

(  g^ =head  lost  =  — .   Assumption  is  made  that  resistance  varies  as  v2.  \ 


CHAPTER  XII. 

RESISTANCE   TO   THE   MOTION  OF  BODIES  IN  WATER. 

281.    Froude's*   experiments    to    determine   frictional  re- 
sistances of  thin  boards  when  propelled  in  water. 

It  has  been  shown  that  the  frictional  resistance  to  the  flow  of 
water  along  pipes  is  proportional  to  the  velocity  raised  to  some 
power  n,  which  approximates  to  two,  and  Mr  Froude's  classical 
experiments,  in  connection  with  the  resistance  of  ships,  show  that 
the  resistance  to  motion  of  plane  vertical  boards  when  propelled 
in  water,  follows  a  similar  law. 


Fig.  362. 

The  experiments  were  carried  out  near  Torquay  in  a  parallel 
sided  tank  278  feet  long,  36  feet  broad  and  10  feet  deep.  A  light 
railway  on  "which  ran  a  stout  framed  truck,  suspended  from  the 
axles  of  two  pairs  of  wheels,"  traversed  the  whole  length  of  the 
tank,  about  20  inches  above  the  water  level.  The  truck  was  pro- 
pelled by  an  endless  wire  rope  wound  on  to  a  barrel,  which  could 
be  made  to  revolve  at  varying  speeds,  so  that  the  truck  could 
traverse  the  length  of  the  tank  at  any  desired  velocity  between 
100  and  1000  feet  per  minute. 

*  Brit.  Ass.  Reports,  1872-4. 


508  HYDRAULICS 

Planes  of  wood,  about  T3F  inch  thick,  the  surfaces  of  which  were 
covered  with  various  materials  as  set  out  in  Table  XXXIX,  were 
made  of  a  uniform  depth  of  19  inches,  and  when  under  experi- 
ment were  placed  on  edge  in  the  water,  the  upper  edge  being 
about  1|  inches  below  the  surface.  The  lengths  were  varied  from 
2  to  50  feet. 

The  apparatus  as  used  by  Froude  is  illustrated  and  described 
in  the  British  Association  Reports  for  1872. 

A  later  adaptation  of  the  apparatus  as  used  at  Haslar  for 
determining  the  resistance  of  ships'  models  is  shown  in  Fig.  362. 
An  arm  L  is  connected  to  the  model  and  to  a  frame  beam,  which 
is  carried  on  a  double  knife  edge  at  H.  A  spring  S  is  attached  to 
a  knife  edge  on  the  beam  and  to  a  fixed  knife  edge  N  on  the 
frame  of  the  truck.  A  link  J  connects  the  upper  end  of  the  beam 
to  a  multiplying  lever  which  moves  a  pen  D  over  a  recording 
cylinder.  This  cylinder  is  made  to  revolve  by  means  of  a  worm 
and  wheel,  the  worm  being  driven  by  an  endless  belt  from  the  axle 
of  the  truck.  The  extension  of  the  spring  S  and  thus  the  move- 
ment of  the  pen  D  is  proportional  to  the  resistance  of  the  model, 
and  the  rotation  of  the  drum  is  proportional  to  the  distance  moved. 
A  pen  A  actuated  by  clockwork  registers  time  on  the  cylinder. 
The  time  taken  by  the  truck  to  move  through  a  given  distance 
can  thus  be  determined. 

To  calibrate  the  spring  S,  weights  W  are  hung  from  a  knife 
edge,  which  is  exactly  at  the  same  distance  from  H  as  the  points 
of  attachment  of  L  and  the  spring  S. 

From  the  results  of  these  experiments,  Mr  Froude  made  the 
following  deductions. 

(1)  The  frictional    resistance   varies  very  nearly  with    the 
square  of  the  velocity. 

(2)  The  mean  resistance  per  square  foot  of  surface  for  lengths 
up  to  50  feet  diminishes  as  the  length  is  increased,  but  is  prac- 
tically constant  for  lengths  greater  than  50  feet. 

(3)  The  frictional  resistance  varies  very  considerably  with 
the  roughness  of  the  surface. 

Expressed  algebraically  the  frictional  resistance  to  the  motion 
of  a  plane  surface  of  area  A  when  moving  with  a  velocity  v  feet 
per  second  is 

_/Q.Ai?» 

^ 


/  being  equal  to 


RESISTANCE   TO   THE    MOTION    OF   BODIES    IN    WATER 


509 


TABLE  XXXIX. 

Showing  the  result  of  Mr  Fronde's  experiments  on  the  frictional 
resistance  to  the  motion  of  thin  vertical  boards  towed  through 
water  in  a  direction  parallel  to  its  plane. 

Width  of  boards  19  inches,  thickness  f\  inch. 

n  =  power  or  index  of  speed  to  which  resistance  is  approxi- 
mately proportional. 

/0  =  the  mean  resistance  in  pounds  per  square  foot  of  a  surface, 
the  length  of  which  is  that  specified  in  the  heading,  when  the 
velocity  is  10  feet  per  second. 

/i  =  the  resistance  per  square  foot,  at  a  distance  from  the 
leading  edge  of  the  board,  equal  to  that  specified  in  the  heading, 
at  a  velocity  of  10  feet  per  second. 

As  an  example,  the  resistance  of  the  tinfoil  surface  per  square 
foot  at  8  feet  from  the  leading  edge  of  the  board,  at  10  feet  per 
second,  is  estimated  at  0*263  pound  per  square  foot;  the  mean 
resistance  is  0'278  pound  per  square  foot. 


Length  of  planes 

2  feet 

8  feet 

20  feet 

50  feet 

Surface 
covered  with 

n 

f. 

/i 

n 

/o 

/i 

n 

/o 

/, 

n 

/o 

/i 

Varnish 

2-0 

0-41 

0-390 

1-85 

0-325 

0-264 

1-85 

0-278 

0-240 

1-83 

0-250 

0-226 

Tinfoil 

2-16 

0-30 

0-295 

1-99 

0-278 

0-263 

1-90 

0-262 

0-244 

1-83 

0-246 

0-232 

Calico 

1-93 

0-87 

0-725 

1-92 

0-626 

0-504 

1-89 

0-531 

0-447 

1-87 

0-474 

0-423 

Fine  sand 

2-0 

0-81 

0-690 

2-0 

0-583 

0-450 

2-0 

0-480 

0-384 

2-06 

0-405 

0-337 

Medium  sand 

2-0 

0-90 

0-730 

2-0 

0-625 

0-488 

2-0 

0-534 

0-465 

2-00 

0-488 

0-456 

Coarse  sand 

2-0 

1-10 

0-880 

2-0 

0-714 

0-520 

2-0 

0-588 

0-490 

The  diminution  of  the  resistance  per  unit  area,  with  the  length, 
is  principally  due  to  the  relative  velocity  of  the  water  and  the 
board  not  being  constant  throughout  the  whole  length. 

As  the  board  moves  through  the  water  the  frictional  resistance 
of  the  first  foot  length,  say,  of  the  board,  imparts  momentum  to 
the  water  in  contact  with  it,  and  the  water  is  given  a  velocity  in 
the  direction  of  motion  of  the  board.  The  second  foot  length  will 
therefore  be  rubbing  against  water  having  a  velocity  in  its  own 
direction,  and  the  frictional  resistance  will  be  less  than  for  the 
first  foot.  The  momentum  imparted  to  the  water  up  to  a  certain 
point,  is  accumulative,  and  the  total  resistance  does  not  therefore 
increase  proportionally  with  the  length  of  the  board. 


510 


HYDRAULICS 


282.  Stream  line  theory  of  the  resistance  offered  to  the 
motion  of  bodies  in  water. 

Resistance  of  ships.  In  considering  the  motion  of  water  along 
pipes  and  channels  of  uniform  section,  the  water  has  been  assumed 
to  move  in  "  stream  lines,"  which  have  a  relative  motion  to  the 
sides  of  the  pipe  or  channel  and  to  each  other,  and  the  resistance 
to  the  motion  of  the  water  has  been  considered  as  due  to  the 
friction  between  the  consecutive  stream  lines,  and  between  the 
water  and  the  surface  of  the  channel,  these  frictional  resistances 
above  certain  speeds  being  such  as  to  cause  rotational  motions  in 
the  mass  of  the  water. 


Fig.  363. 


Fig.  364. 

It  has  also  been  shown  that  at  any  sudden  enlargement  of  a 
stream,  energy  is  lost  due  to  eddy  motions,  and  if  bodies,  such 
as  are  shown  in  Figs.  110  and  111,  be  placed  in  the  pipe,  there  is 
a  pressure  acting  on  the  body  in  the  direction  of  motion  of  the 
water.  The  origin  of  the  resistance  of  ships  is  best  realised  by 
the  "stream  line"  theory,  in  which  it  is  assumed  that  relative  to 
the  ship  the  water  is  moving  in  stream  lines  as  shown  in  Figs. 
363,  364,  consecutive  stream  lines  also  having  relative  motion. 


RESISTANCE  TO   THE   MOTION   OF   BODIES   IN   WATER          511 

According  to  this  theory  the  resistance  is  divided  into  three 
parts. 

(1)  Frictional  resistance  due  to  the  relative  motions  of  con- 
secutive stream  lines,  and  of  the  stream  lines  and  the   surface 
of  the  ship. 

(2)  Eddy  motion  resistances  due  to  the   dissipation   of  the 
energy  of  the  stream  lines,  all  of  which  are  not  gradually  brought 
to  rest. 

(3)  Wave  making  resistances  due  to  wave  motions  set  up  at 
the  surface  of  the  water  by  the  ship,  the  energy  of  the  waves 
being  dissipated  in  the  surrounding  water. 

According  to  the  late  Mr  Froude,  the  greater  proportion  of 
the  resistance  is  due  to  friction,  and  especially  is  this  so  in  long 
ships,  with  fine  lines,  that  is  the  cross  section  varies  very  gradually 
from  the  bow  towards  midships,  and  again  from  the  midships 
towards  the  stern.  At  speeds  less  than  8  knots,  Mr  Froude  has 
shown  that  the  frictional  resistance  of  ships,  the  full  speed  of 
which  is  about  13  knots,  is  nearly  90  per  cent,  of  the  whole 
resistance,  and  at  full  speed  it  is  not  much  less  than  60  per  cent. 
He  has  further  shown  that  it  is  practically  the  same  as  that 
resisting  the  motion  of  a  thin  rectangle,  the  length  and  area  of 
the  two  sides  of  which  are  equal  to  the  length  and  immersed 
area  respectively  of  the  ship,  and  the  surface  of  which  has  the 
same  degree  of  roughness  as  that  of  the  ship. 

If  A  is  the  area  of  the  immersed  surface,  /  the  coefficient  of 
friction,  which  depends  not  only  upon  the  roughness  but  also 
upon  the  length,  V  the  velocity  of  the  ship  in  feet  per  second,  the 
resistance  due  to  friction  is 

r,=/.A.V", 

the  value  of  the  index  n  approximating  to  2. 

The  eddy  resistance  depends  upon  the  bluntness  of  the  stern  of 
the  boat,  and  can  be  reduced  to  a  minimum  by  diminishing  the 
section  of  the  ship  gradually,  as  the  stern  is  approached,  and  by 
avoiding  a  thick  stern  and  stern  post. 

As  an  extreme  case  consider  a  ship  of  the  section  shown  in 
Fig.  364,  and  suppose  the  stream  lines  to  be  as  shown  in  the 
figure.  At  the  stern  of  the  boat  a  sudden  enlargement  of  the 
stream  lines  takes  place,  and  the  kinetic  energy,  which  has  been 
given  to  the  stream  lines  by  the  ship,  is  dissipated.  The  case  is 
analogous  to  that  of  the  cylinder,  Fig.  Ill,  p.  169.  Due  to  the 
loss  of  energy,  or  head,  there  is  a  resultant  pressure  acting  upon 
the  ship  in  the  direction  of  flow  of  the  stream  lines,  and  con- 
sequently opposing  its  motion. 


512  HYDRAULICS 

If  the  ship  has  fine  lines  towards  the  stern,  as  in  Fig.  363, 
the  velocities  of  the  stream  lines  are  diminished  gradually  and  the 
loss  of  energy  by  eddy  motions  becomes  very  small.  In  actual 
ships  it  is  probably  not  more  than  8  per  cent,  of  the  whole 
resistance. 

The  wave  making  resistance  depends  upon  the  length  and  the 
form  of  the  ship,  and  especially  upon  the  length  of  the  "entrance" 
and  "  run."  By  the  "  entrance  "  is  meant  the  front  part  of  the 
ship,  which  gradually  increases  in  section*  until  the  middle  body, 
which  is  of  uniform  section,  is  reached,  and  by  the  "run,"  the 
hinder  part  of  the  ship,  which  diminishes  in  section  from  the 
middle  body  to  the  stern  post. 

Beyond  a  certain  speed,  called  the  critical  speed,  the  rate  of 
increase  in  wave  making  resistance  is  very  much  greater  than 
the  rate  of  increase  of  speed.  Mr  Froude  found  that  for  the 
S.S.  "Merkara"  the  wave  making  resistance  at  13  knots,  the 
normal  speed  of  the  ship,  was  17  per  cent,  of  the  whole,  but  at  19 
knots  it  was  60  per  cent.  The  critical  speed  was  about  18  knots. 

An  approximate  formula  for  the  critical  speed  Y  in  knots  is 


L  being  the  length  of  entrance,  and  LI  the  length  of  the  run  in 
feet. 

The  mode  of  the  formation  by  the  ship  of  waves  can  be  partly 
realised  as  follows. 

Suppose  the  ship  to  be  moving  in  smooth  water,  and  the  stream 
lines  to  be  passing  the  ship  as  in  Fig.  363.  As  the  bow  of  the 
boat  strikes  the  dead  water  in  front  there  is  an  increase  in 
pressure,  and  in  the  horizontal  plane  SS  the  pressure  will  be 
greater  at  the  bow  than  at  some  distance  in  front  of  it,  and 
consequently  the  water  at  the  bow  is  elevated  above  the  normal 
surface. 

Now  let  AA,  BB,  and  CO  be  three  sections  of  the  ship  and  the 
stream  lines. 

Near  the  midship  section  CO  the  stream  lines  will  be  more 
closely  packed  together,  and  the  velocity  of  flow  will  be  greater, 
therefore,  than  at  AA  or  BB.  Assuming  there  is  no  loss  of  energy 
in  a  stream  line  between  AA  and  BB  and  applying  Bernoulli's 
theorem  to  any  stream  line, 

PA  +  ^A2  =  PC  +  ^  =  £B  +  V 
w      2g      w      2g      w      2g> 

*  See  Sir  W.  White's  Naval  Architecture,  Transactions  of  Naval  Architects, 
1877  and  1881. 


RESISTANCE  TO  THE  MOTION   OF   BODIES   IN   WATER         513 
and  since  VA  and  v^  are  less  than  VG, 

^  and  £?  are  greater  than  ^. 
w  w  w 

The  surface  of  the  water  at  A  A  and  BB  is  therefore  higher 
than  at  CC  and  it  takes  the  form  shown  in  Fig.  363. 

Two  sets  of  waves  are  thus  formed,  one  by  the  advance  of  the 
bow  and  the  other  by  the  stream  lines  at  the  stern,  and  these 
wave  motions  are  transmitted  to  the  surrounding  water,  where 
their  energy  is  dissipated.  This  energy,  as  well  as  that  lost  in 
eddy  motions,  must  of  necessity  have  been  given  to  the  water  by 
the  ship,  and  a  corresponding  amount  of  work  has  to  be  done  by 
the  ship's  propeller.  The  propelling  force  required  to  do  work 
equal  to  the  loss  of  energy  by  eddy  motions  is  the  eddy  resist- 
ance, and  the  force  required  to  do  work  equal  to  the  energy  of 
the  waves  set  up  by  the  ship  is  the  wave  resistance. 

To  reduce  the  wave  resistance  to  a  minimum  the  ship  should 
be  made  very  long,  and  should  have  no  parallel  body,  or  the 
entire  length  of  the  ship  should  be  devoted  to  the  entrance  and 
run.  On  the  other  hand  for  the  frictional  resistance  to  be  small, 
the  area  of  immersion  must  be  small,  so  that  in  any  attempt 
to  design  a  ship  the  resistance  of  which  shall  be  as  small  as 
possible,  two  conflicting  conditions  have  to  be  met,  and,  neglecting 
the  eddy  resistances,  the  problem  resolves  itself  into  making  the 
sum  of  the  frictional  and  wave  resistances  a  minimum. 

Total  resistance.  If  K,  is  the  total  resistance  in  pounds,  rf  the 
frictional  resistance,  re  the  eddy  resistance,  and  rw  the  wave 
resistance, 


The  frictional  resistance  rf  can  easily  be  determined  when  the 
nature  of  the  surface  is  known.  For  painted  steel  ships  /  is 
practically  the  same  as  for  the  varnished  boards,  and  at  10  feet 
per  second  the  frictional  resistance  is  therefore  about  £  Ib.  per 
square  foot,  and  at  20  feet  per  second  1  Ib.  per  square  foot.  The 
only  satisfactory  way  to  determine  re  and  rw  for  any  ship  is  to 
make  experiments  upon  a  model,  from  which,  by  the  principle  of 
similarity,  the  corresponding  resistances  of  the  ship  are  deduced. 
The  horse-power  required  to  drive  the  ship  at  a  velocity  of  V  feet 
per  second  is 

TTP-RV 

=        ' 


To  determine  the  total  resistance  of  the  model  the  apparatus 
shown  in  Fig.  362  is  used  in  the  same  way  as  in  determining  the 
frictional  resistance  of  thin  boards. 

L.  H.  33 


514  HYDRAULICS 

283.  Determination  of  the  resistance  of  a  ship  from  the 
resistance  of  a  model  of  the  ship. 

To  obtain  the  resistance  of  the  ship  from  the  experimental 
resistance  of  the  model  the  principle  of  similarity,  as  stated  by 
Mr  Froude,  is  used.  Let  the  linear  dimensions  of  the  ship  be  D 
times  those  of  the  model. 

Corresponding  speeds.  According  to  Mr  Fronde's  theory,  for 
any  speed  V™  of  the  model,  the  speed  of  the  ship  at  which  its 
resistance  must  be  compared  with  that  of  the  model,  or  the 
corresponding  speed  Ys  of  the  ship,  is 


Corresponding  resistances.  If  Rm  is  the  resistance  of  the  model 
at  the  velocity  Ym,  and  it  be  assumed  that  the  coefficients  of 
friction  for  the  ship  and  the  model  are  the  same,  the  resistance  Rs 
of  the  ship  at  the  corresponding  speed  Vs  is 


As  an  example,  suppose  a  model  one-sixteenth  of  the  size 
of  the  ship  ;  the  corresponding  speed  of  the  ship  will  be  four  times 
the  speed  of  the  model,  and  the  resistance  of  the  ship  at  corre- 
sponding speeds  will  be  163  or  4096  times  the  resistance  of  the 
model. 

Correction  for  the  difference  of  the  coefficients  of  friction  for  the 
model  and  ship.  The  material  of  which  the  immersed  surface  of 
the  model  is  made  is  not  generally  the  same  as  that  of  the  ship, 
and  consequently  R8  must  be  corrected  to  make  allowance  for  the 
difference  of  roughness  of  the  surfaces.  In  addition  the  ship  is 
very  much  longer  than  the  model,  and  the  coefficient  of  friction, 
even  if  the  surfaces  were  of  the  same  degree  of  roughness,  would 
therefore  be  less  than  for  the  model. 

Let  Am  be  the  immersed  surface  of  the  model  and  As  of 
the  ship. 

Let  fm  be  the  coefficient  of  friction  for  the  model  and  fs  for  the 
ship,  the  values  being  made  to  depend  not  only  upon  the  roughness 
but  also  upon  the  length.  If  the  resistance  is  assumed  to  vary  as 
V2,  the  frictional  resistance  of  the  model  at  the  velocity  Vm  is 

r    —  f  A    V  2 

'  m       J  m-E*-m,  "  m  j 

and  for  the  ship  at  the  corresponding   speed  Vs  the  frictional 
resistance  is 


But  As-AmD2 


RESISTANCE  TO   THE   MOTION   OF   BODIES   IN  WATER 

and,  therefore,  r»  =  /»AmVm2  D3 

=  £rJD3. 

Jm 

Then  the  resistance  of  the  ship  is 


515 


Determination  of  the  curve  of  resistance  of  the  ship  from  the 
curve  of  resistance  of  the  model.  From  the  experiments  on  the 
model  a  curve  having  resistances  as  ordinates  and  velocities  as 
abscissae  is  drawn  as  in  Fig.  365.  If  now  the  coefficients  of 
friction  for  the  ship  and  the  model  are  the  same,  this  curve,  by 
an  alteration  of  the  scales,  becomes  a  curve  of  resistance  for  the 
ship. 

For  example,  in  the  figure  the  dimensions  of  the  ship  are 
supposed  to  be  sixteen  times  those  of  the  model.  The  scale  of 
velocities  for  the  ship  is  shown  on  CD,  corresponding  velocities 
being  four  times  as  great  as  the  velocity  of  the  model,  and  the 
scale  of  resistances  for  the  ship  is  shown  at  EH,  corresponding 
resistances  being  4096  times  the  resistance  of  the  model. 


H 


JE 


C  D 

Fig.  365. 

Mr  Froude's  method  of  correcting  the  curve  for  the  difference  of 
the  coefficients  of  friction  for  the  ship  and  the  model.  From  the 
formula 

rm  =  fmA.mVmn, 

33—2 


516  HYDRAULICS 

the  frictional  resistance  of  .the  model  for  several  values  of  Vm 
is  calculated,  and  the  curve  FF  plotted  on  the  same  scale  as  used 
for  the  curve  RE,.  The  wave  and  eddy  making  resistance  at  any 
velocity  is  the  ordinate  between  FF  and  RE.  At  velocities  of 
200  feet  per  second  for  the  model  and  800  feet  per  second  for 
the  ship,  for  example,  the  wave  and  eddy  making  resistance  is  fee, 
measured  on  the  scale  BGr  for  the  model  and  on  the  scale  EH  for 
the  ship. 

The  frictional  resistance  of  the  ship  is  now  calculated  from  the 
formula  rs=f8AsV8n,  and  ordinates  are  set  down  from  the  curve 
FF,  equal  to  rs,  to  the  scale  for  ship  resistance.  A  third  curve  is 
thus  obtained,  and  at  any  velocity  the  ordinate  between  this  curve 
and  EE  is  the  resistance  of  the  ship  at  that  velocity.  For  example, 
when  the  ship  has  a  velocity  of  800  feet  per  second  the  resistance 
is  ac,  measured  on  the  scale  EH. 


EXAMPLES. 

(1)  Taking  skin  friction  to  be  0'4  Ib.  per  square  foot  at  10  feet  per 
second,  find  the  skin  resistance  of  a  ship  of  12,000  square  feet  immersed 
surface  at  15  knots  (1  knot  =  T69  feet  per  second).    Also  find  the  horse-power 
to  drive  the  ship  against  this  resistance. 

(2)  If  the  skin  friction  of  a  ship  is  0*5  of  a  pound  per  square  foot  of 
immersed  surface  at  a  speed  of  6  knots,  what  horse-power  will  probably 
be  required  to  obtain  a  speed  of  14  knots,  if  the  immersed  surface  is  18,000 
square  feet  ?    You  may  assume  the  maximum  speed  for  which  the  ship  is 
designed  is  17  knots. 

(3)  The  resistance  of  a  vessel  is  deduced  from  that  of  a  model  j^th  the 
linear  size.     The  wetted  surface  of  the  model  is  29*4  square  feet,  the  skin 
friction  per  square  foot,  in  fresh  water,  at  10  feet  per  second  is  0'3  Ib.,  and 
the  index  of  velocity  is  1'94.     The  skin  friction  of  the  vessel  in  salt  water 
is  60  Ibs.  per  100  square  feet  at  10  knots,  and  the  index  of  velocity  is  T83. 
The  total  resistance  of  the  model  in  fresh  water  at  200  feet  per  minute  is 
1-46  Ibs.    Estimate  the  total  resistance  of  the  vessel  in  salt  water  at  the 
speed  corresponding  to  200  feet  per  minute  in  the  model.    Lond.  Un.  1906. 

(4)  How  from  model  experiments  may  the  resistance  of  a  ship  be 
inferred?    Point  out  what  corrections  have  to  be  made.     At  a  speed  of 
300  feet  per  minute  in  fresh  water,  a  model  10  feet  in  length  with  a  wet 
skin  of  24  square  feet  has  a  total  resistance  of  2'39  Ibs.,  2  Ibs.  being  due  to 
skin  resistance,  and  -39  Ib.  to  wave-making.     What  will  be  the  total  resist- 
ance at  the  corresponding  speed  in  salt  water  of  a  ship  25  times  the  linear 
dimensions  of  the  model,  having  given  that  the  surface  friction  per  square 
foot  of  the  ship  at  that  speed  is  1-3  Ibs.  ?     Lond.  Un.  1906. 


CHAPTER  XIII. 

STREAM  LINE   MOTION. 

284.  Hele  Shaw's  experiments  on  the  flow  of  thin 
sheets  of  water. 

Professor  Hele  Shaw*  has  very  beautifully  shown,  on  a  small 
scale,  the  form  of  the  stream  lines  in  moving  masses  of  water 
under  varying  circumstances,  and  has  exhibited  the  change  from 
stream  line  to  sinuous,  or  rotational  flow,  by  experiments  on  the 
flow  of  water  at  varying  velocities  between  two  parallel  glass 
plates.  In  some  of  the  experiments  obstacles  of  various  forms 
were  placed  between  the  plates,  past  which  the  water  had  to  flow, 
and  in  others,  channels  of  various  sections  were  formed  through 
which  the  water  was  made  to  flow.  The  condition  of  the  water 
as  it  flowed  between  the  plates  was  made  visible  by  mixing  with 
it  a  certain  quantity  of  air,  or  else  by  allowing  thin  streams  of 
coloured  water  to  flow  between  the  plates  along  with  the  other 
water.  When  the  velocity  of  flow  was  kept  sufficiently  low, 
whatever  the  form  of  the  obstacle  in  the  path  of  the  water,  or 
the  form  of  the  channel  along  which  it  flowed,  the  water  persisted 
in  stream  line  flow.  When  the  channel  between  the  plates  was 
made  to  enlarge  suddenly,  as  in  Fig.  58,  or  to  pass  through  an 
orifice,  as  in  Fig.  59,  and  as  long  as  the  flow  was  in  stream  lines, 
no  eddy  motions  were  produced  and  there  were  no  indications 
of  loss  of  head.  When  the  velocity  was  sufficiently  high  for  the 
flow  to  become  sinuous,  the  eddy  motions  were  very  marked. 
When  the  motion  was  sinuous  and  the  water  was  made  to  flow 
past  obstacles  similar  to  those  indicated  in  Figs.  110  and  111,  the 
water  immediately  in  contact  with  the  down-stream  face  was 
shown  to  be  at  rest.  Similarly  the  water  in  contact  with  the 
annular  ring  surrounding  a  sudden  enlargement  appeared  to  be 
at  rest  and  the  assumption  made  in  section  51  was  thus  justified. 

*  Proceedings  of  Naval  Architects,  1897  and  1898.     Engineer,  Aug.  1897  and 
May  1898. 


518  HYDRAULICS 

When  the  flow  was  along  channels  and  sinuous,  the  sinuously 
moving  water  appeared  to  be  separated  from  the  sides  of  the 
channel  by  a  thin  film  of  water,  which  Professor  Hele  Shaw 
suggested  was  moving  in  stream  lines,  the  velocity  of  which  in 
the  film  diminish  as  the  surface  of  the  channel  is  approached. 
The  experiments  also  indicated  that  a  similar  film  surrounded 
obstacles  of  ship-like  and  other  forms  placed  in  flowing  water, 
and  it  was  inferred  by  Professor  Hele  Shaw  that,  surrounding 
a  ship  as  it  moves  through  still  water,  there  is  a  thin  film  moving 
in  stream  lines  relatively  to  the  ship,  the  shearing  forces  between 
which  and  the  surrounding  water  set  up  eddy  motions  which 
account  for  the  skin  friction  of  the  ship. 

285.     Curved  stream  line  motion. 

Let  a  mass  of  fluid  be  moving  in  curved  stream  lines,  and  let 
AB,  Fig.  366,  be  any  one  of  the  stream  lines. 

At  any  point  c  let  the  radius  of  curvature  of  the  stream  line 
be  r  and  let  0  be  the  centre  of  curvature. 

Consider  the  equilibrium  of  an  element  dbde  surrounding  the 
point  c. 

Let  W  be  the  weight  of  this  element. 

p  be  the  pressure  per  unit  area  on  the  face  bd. 

p  +  dploe  the  pressure  per  unit  area  on  the  face  ae. 

0  be  the  inclination  of  the  tangent  to  the  stream  line  at  c 

to  the  horizontal. 

a  be  the  area  of  each  of  the  faces  bd  and  ae. 
v  be  the  velocity  of  the  stream  line  at  c. 
dr  be  the  thickness  ab  of  the  stream  line. 

If  then  the  stream  line  is  in  a  vertical  plane  the  forces  acting 
on  the  element  are 

(1)  W  due  to  gravity, 

-TTTT-    2 

(2)  the  centrifugal  force  — —  acting  along  the  radius  away 
from  the  centre,  and 

(3)  the  pressure  adp  acting  along  the   radius  towards  the 
centre  of  curvature  0. 

Resolving  along  the  radius  through  c, 

adp +  W  cos  0  =  0, 

gr 

or  since  W  =  wadr, 

dp     wv2 
dr=^~ 
If  the  stream  line  is  horizontal,  as  in  the  case  of  water  flowing 


STREAM   LINE   MOTION 


519 


round  the  bend  of  a  river,  Oc  is  horizontal  and  the  component  of 
W  along  Oc  is  zero. 

Then  f=™V-  ...(2). 

dr      g  r 

Integrating  between  the  limits   R  and   Ra  the  difference  of 
pressure  on  any  horizontal  plane  at  the  radii  R  and  R!  is 


W    [ R'  V*    , 

Pi  —  p  =  —         —dr 

g  J  R  r 


(3), 


which  can  be  integrated  when  v  can  be  written  as  a  function  of  r. 
Now  for  any  horizontal   stream    line,   applying    Bernoulli's 
equation, 


. 
-  is  constant, 


or 


Differentiating 


!_  dp     vdv  _  dK 

w  dr     gdr      dr     


Fig.  366. 


Fig.  367. 


Free  vortex.  An  important  case  arises  when  H  is  constant  for 
all  the  stream  lines,  as  when  water  flows  round  a  river  bend,  or  as 
in  Thomson's  vortex  chamber. 


Then 


1  dp_  -vdv 
w  dr~    gdr 
dp 


.(5). 


Substituting  the  value  of  -gf  from  (5)  in  (2) 

—  wv  dv  _w   tf 
g     dr  ~  g  '  r  ' 

from  which  rdv  +  vdr  =  0, 

and  therefore  by  integration 

vr  =  constant  =  C 


520  HYDRAULICS 

Equation  (3)  now  becomes 

gh-p  =  C2  /*'  dr 
w          g  JR   rs 


_ 

-20VK2 

Forced  vortex.  If,  as  in  the  turbine  wheel  and  centrifugal 
pump,  the  angular  velocities  of  all  the  stream  lines  are  the  same, 
then  in  equation  (3) 


and 

w2 

=  vRl~ 

Scouring  of  the  banks  of  a  river  at  the  bends.  When  water 
runs  round  a  bend  in  a  river  the  stream  lines  are  practically 
concentric  circles,  and  since  at  a  little  distance  from  the  bend  the 
surface  of  the  water  is  horizontal,  the  head  H  on  any  horizontal 
in  the  bend  must  be  constant,  and  the  stream  lines  form  a  free 
vortex.  The  velocity  of  the  outer  stream  lines  is  therefore  less 
than  the  inner,  while  the  pressure  head  increases  as  the  outer 
bank  is  approached,  and  the  water  is  consequently  heaped  up 
towards  the  outer  bank.  The  velocity  being  greater  at  the  inner 
bank  it  might  be  expected  that  it  will  be  scoured  to  a  greater 
extent  than  the  outer.  Experience  shows  that  the  opposite  effect 
takes  place.  Near  the  bed  of  the  river  the  stream  lines  have  a 
less  velocity  (see  page  209)  than  in  the  mass  of  the  fluid,  and,  as 
Lord  Kelvin  has  pointed  out,  the  rate  of  increase  of  pressure  near 
the  bed  of  the  stream,  due  to  the  centrifugal  forces,  will  be  less 
than  near  the  surface.  The  pressure  head  near  the  bed  of  the 
stream,  due  to  the  centrifugal  forces,  is  thus  less  than  near  the 
surface,  and  this  pressure  head  is  consequently  unable  to  balance 
the  pressure  head  due  to  the  heaping  of  the  surface  water,  and 
cross-currents  are  set  up,  as  indicated  in  Fig.  367,  which  cause 
scouring  of  the  outer  bank  and  deposition  at  the  inner  bank. 


ANSWERS   TO   EXAMPLES. 


Chapter  I. 

(1)  3900  Ibs.    9372  Ibs.            (2)  784  Ibs.            (3)     78'6  tons. 

(4)  5880  Ibs.            (5)     17'1  feet.  (6)     19800  Ibs. 

(7)  P  =  665,600  Ibs.     X  =  12'5  ft.  (8)     '91  foot.             (9)     '089  in. 

(10)  15-95  Ibs.  per  sq.  ft.             (11)  5400  Ibs.             (12)     87040  Ibs. 

Chapter  II. 

(1)     35,000  c.  ft.  (3)     2-98  ft. 

(4)     Depth  of  C.  of  B.  =  21-95  ft.     BM  =  14-48  ft.  (5)     19'1  ft.     6'9  ft. 

(6)  Less  than  13'8  ft.  from  the  bottom.  (7)     1'57  ft.  (8)     2'8  ins. 

Chapter  III. 

(1)  -945.  (2)     14-6  ft.  per  sec.     18'3  c.  ft.  per  sec.  (3)     25 '01  ft. 

(4)  115  ft.  (5)     53-3  ft.  per  sec.  (6)     63  c.  ft.  per  sec. 

(7)  44928  ft.  Ibs.     T36  H.P.     8'84  ft.  (8)     86'2  ft.     11-4  ft.  per  sec. 
(9)  1048  gallons. 

Chapter  IV,  page  78. 

(1)     80-25.  (2)     3906.  (3)     37'636.  (4)     5  ins.  diam. 

(5)  3-567  ins.  (6)     -763.  (7)     86  ft.  per  sec.     115ft. 

(8)  -806.  (9)     -895.  (10)     -058.  (11)     144-3  ft.  per  sec. 

(12)  2-94  ins.  (13)     -60.  (14)     572  gallons.  (15)     22464  Ibs. 
(16)     -6206.             (17)     5-53  c.  ft.             (18)     '755.             (19)     102  c.  ft. 

(20)  -875  ft.     136  Ibs.  per  sq.  foot.     545  ft.  Ibs. 

(21)  10-5  ins.     29-85  ins.  (22)     -683  ft.  (23)     4-52  minutes. 
(24)     17-25  minutes.             (25)     -629  sq.  ft.             (26)     1'42  hours. 

Chapter  IV,  page  110. 

(1)  13,170  c.  ft.  (2)  4-15  ft. 

(8)  69-9  c.  ft.  per  sec.  129-8  c.  ft.  per  sec.       (4)  2-635. 
(5)  13-28.       (7)  43-3  c.  ft.  per  see.       (8)  1-675  ft. 

(9)  89-2  ft.     (10)  2-22  ft.     (11)  5 -52  ft.      (12)  23,500  c.  ft. 

(13)  24,250  c.  ft.  (14)     105  minutes.  (15)     284  H.P. 


522  ANSWERS   TO   EXAMPLES 


Chapter  V. 

(1)  27'8  ft.  (2)     142  ft.  (4)     '65.  (5)     2'388  ft. 

(6)  10-75.     1-4  ft.     '33ft.     -782ft.     -0961ft. 

(8)  -61  c.  ft.     28-54  ft.     25-8  ft.     9  ft.  (9)     26  per  cent. 

(10)  1-97.     21ft.     30ft.     26ft.     24ft.     15ft.  (11)     3'64  c.  ft. 

(12)  3-08  c.  ft.  (13)     -574  ft.     -267  ft.     7'72  ft.  (14)     2'1  ft. 

(15)  1-86  c.  ft.  per  sec.  (16)     F  =  '0318lbs.    /= -005368. 

(17)  1-023.  (18)     -704.  (19)     2-9  ft.  per  sec. 

(20)  4-4  c.  ft.  per  sec.  (21)     If  pipe  is  clean  46  ft. 

(22)  23  ft.     736  ft.  (23)     Dirty  cast-iron  6'1  feet  per  mile. 

(24)  8-18  feet.  (26)     1  foot. 

(27)  ~TrT~  '  F  =  friction  per 'unit  area  at  unit  velocity. 

(28)  108  H.  P.  (29)     1480  Ibs.     1 -03  ins.  (30)     -002825. 

(31)  k= -004286.  w=l-84.       (32)    (a)   940ft.    (b)   2871  H.  P.        (33)    -0458, 

(34)  If  d=9",  v=5  ft.  per  sec.,  and  /=  -0056, 7i  =  102  and  H  =  182. 

(35)  1487  xlO4.     Yes.  (36)     58-15  ft.  (37)     1  hour  48  min. 
(38)  46,250  gallons.     Increase  17  per  cent.  (39)     295'7  feet. 
(40)  6  pipes.     480  Ibs.  per  sq.  inch. 

(42)     Velocities  6'18,  5-08,  8-15  ft.  per  sec.    Quantity  to  B=60  c.  ft.  per  min. 
Quantity  to  C=66'6  c.  ft.  per  min.  (45)     -468  c.  ft.  per  sec. 

(46)  Using  formula  for  old  cast-iron  pipes  from  page  138,  ?;  =  3-62  ft.  per  sec. 

(47)  2-91  ft.  (48)     d= 3'8  ins.     ^  =  3-4  ins.     d2  =  2'9  ins.    d3  =  2'2  ins. 
(49)     Taking  C  as  120,  first  approximation  to  Q  is  14-4  c.  ft.  per  sec. 

(51)     d=4-13  ins.     v  =  20'55  ft.  per  sec.    _p  =  840  Ibs.  per  sq.  inch. 

(53)  7-069  ft.    3-01  ft.     Cr=ll-9.     Cr  for  tubes  =  5 -06. 

(54)  Loss  of  head  by  friction  =  -73  ft. 

v2 
A  head  equal  to  ^-  will  probably  be  lost  at  each  bend. 

(56)  43-9  ft.     -936  in. 

(57)  h = 58'.     Taking  -005  to  be  /  in  formula  h  =  ^? ,  v  =  16'6  ft.  per  sec. 

(58)  ^  =  8-8  ft.  per  sec.  from  A  to  P.     v2  =  4-95  ft.  per  sec.  from  B  to  P. 

V3= 13-75  ft.  per  sec.  from  P  to  C. 


Chapter  VI. 

(!)     88-5.  (2)     1-1  ft.  diam. 

(3)  Value  of  ra  when  discharge  is  a  maximum  is  1-357.   o>  =  17*62.   C  =  127, 

Q  =  75  c.  ft.  per  sec. 

(4)  -0136.  (5)     16,250  c.  ft.  per  sec.  (6)     3ft. 

(7)  Bottom  width  15  ft.  nearly.  (8)     Bottom  width  10  ft.  nearly. 

(9)  630  c.  ft.  per  sec.  (10)     96,000  c.  ft.  per  sec. 

(11)  Depth  7-35  ft.  (12)     Depth  10'7  ft. 

(13)  Bottom  width  75  ft.     Slope  -00052.  (17)     C  =  87'5. 


ANSWERS  TO   EXAMPLES  52S 


Chapter  VIII. 

(1)  124-8  Ibs.     -456  H.  P.  (2)     623  Ibs. 

(3)  104  Ibs.     58-7  Ibs.     294  ft.  Ibs.  (4)     960  Ibs. 

(5)  261  Ibs.    4-7  H.  P.  (6)     21-8.  (7)     57  Ibs.  (8)     194  Ibs. 

(9)  Impressed  velocity  =  28-5  ft.  per  sec.     Angle  =  57°.  (10)     131  Ibs. 

(12)  -93.     -678.     -63.  (13)     19'2. 

(14)  Vel.  into  tank  =  34-8  ft.  per  sec.     Vel.  through  the  orifice  =  41-5  ft.  per 

sec.     Wt.  lifted  =  10-3  tons.     Increased  resistance =2330  Ibs. 

(15)  125  Ibs.     8-4  ft.  per  sec.     1-91  H.P. 

(16)  Work  done,  575,  970,  1150,  1940  ft.  Ibs.     Efficiencies  fy,  '50,  £f ,  1. 

(17)  1420  H.P.  (18)     -9375.  (19)     32  H.P. 
(20)  3666  Ibs.     161  H.P.     62  per  cent. 

Chapter  IX. 

(1)  105  H.P.  (2)     0=29°.    Vr= 7  ft.  per  sec. 

(3)  14-3  per  min.     11°  from  the  top  of  wheel.    $  =  70°. 

(4)  1-17  c.  ft.  (5)    4-14  ft.  (8)     32°  12'. 
(9)  10-25  ft.  per  sec.     1-7  ft.     5'3  ft.  per  sec.     11°  to  radius. 

(12)  u  =  24-7  ft.  per  sec.  (13)     0  =  47°  30'.     a  =  27°  20'. 

(14)  79°  15'.     19°  26'.     -53. 

(15)  35-6  ft.  per  sec.     6°  24'.     23|  ins.     llf  ins.     12°  39'.     16|ins.    32|ins. 

(16)  99  per  cent.     0=73°,  a  =  18°.    $  =  120°,  a  =  18°. 

(18)  0  =  153°  23'.     H  =  77-64  ft.     H.P.  =  141-16.     Pressure  head  =  48'53  ft. 

(19)  d  =  1-22  ft.     D  =  2-14  ft.     Angles  12°  45',  125°  22',  16°  4'. 

(20)  0  =  134°  53',  6  =  16°  25',  a  =  9°  10'.     H.  p.  =  2760. 

(21)  616.    Heads  by  gauge,  - 14,  35'6,  81.     U  =  51'5  ft.  per  sec. 

(22)  0  =  153°  53',  a  =  25°.     H.  p.  =  29'3.     Eff .  =  "957. 

(23)  Blade  angle  13°  30'.     Vane  angle  30°  25'.     3'92  ft.  Ibs.  per  Ib. 

(24)  At  2'  6"  radius,  6  =  10°,  0  =  23°  45',  a  =  16°  24'.  At  3'  3"  radius,  6  =  12°  11' 

0  =  78°  47',  a  =  12°  45'.   At  4'  radius,  6  =  15°  46',  0  =  152°  11',  a  =  10°  21 . 

(25)  79°  30'.     21°  40'.     41°  30'. 

(26)  53°  40'.    36°.     24°.    86'8  per  cent.    87  per  cent. 

(27)  12°  45'.     62°  15'.     31°  45'. 

(28)  v  =  45-35.    U=77.    Vr=44.    vr  =  36.    Ux  =  23.    e  =  73'75  per  cent. 

(29)  -36ft.    40°  to  radius.  (30)     About  22  ft. 
(31)  H.P. =80-8.     Eff.  =  92-5  per  cent. 


Chapter  X. 

(1)     47-4  H.P.  (2)     25°.     53-1  ft.  per  sec.     94ft.     50ft. 

(3)     55  per  cent.  (4)     52*5  per  cent. 

(5)  ^  =  106  ft.     ^L2=51ft.    ^1  =  55  ft. 

g  2g  w 

(6)  11°  36'.     105ft.     47-4  ft. 

(7)  60  per  cent.     151  H.P.     197  revs,  per  min. 

(8)  700  revs,  per  min.     -81  in.     Radial  velocity  14'2  ft.  per  sec. 
(12)  15-6  ft.  Ibs.  per  Ib.     3 -05  ft.     14  ft.  per  sec. 


524  ANSWERS  TO   EXAMPLES 

(15)  v  =  23-64  ft.  per  sec.    V  =  11'3. 

(16)  d=9£ins.     D  =  19  ins.     Kevs.  per  min.  472  or  higher. 

(17)  15  H.  P.    9-6  ins.  diam.  (18)     5'5  ft. 

(19)  Vels.  1-23  and  2-41  ft.  per  sec.    Max.  accel.  2-32  and  4-55  ft.  per  sec. 

per  sec. 

(20)  393  ft.  Ibs.    Mean  friction  head  =  '0268,  therefore  work  due  to  friction 

is  very  small. 

(21)  4-61  H.  P.     11-91  c.  ft.  per  min.  (22)     -338. 

(23)    p—  —  ^    1.    Acceleration  is  zero  when  0=j(ra-{-2),  m  being  any 

integer. 

(27)     Separation  in  second  case. 

(29)     67'6  and  66-1  Ibs.  per  sq.  inch  respectively.     H.p.  =  3'14. 
(31)     7'93  ft.     25'3  ft.     41 '93  ft.  (32)     '643.  (33)     '6. 

(34)     Separation  in  the  sloping  pipe. 


Chapter  XI. 

(1)     3150  Ibs.  (2)     3-38  H.P.  hours.  (5)     4'7  ins.  and  9'7  ins. 

(6)     3-338  tons.  (7)     175  Ibs.  per  sq.  inch. 

(8)    2-8  ft.  per  sec.  (9)     4-2  minutes. 


Chapter  XII. 

(1)  30,890  Ibs.  1425  H.  P.          (2)  3500  H.  P. 
(3)  4575  Ibs.  (4)  25,650  Ibs. 


INDEX. 


[All  numbers  refer  to  pages.] 


Absolute  velocity  262 

Acceleration    in   pumps,   effect  of   (see 

Reciprocating  pump) 
Accumulators 

air   491 

differential  489 

hydraulic  486 
Air  gauge,  inverted   9 
Air  vessels  on  pumps   451,  455 
Angular  momentum   273 
Angular  momentum,  rate  of  change  of 

equal  to  a  couple   274 
Appold  centrifugal  pump   415 
Aqueducts   1,  189,   195 

sections  of  216 
Archimedes,  principle  of  22 
Armstrong  double  power  hydraulic  crane 

497 
Atmospheric  pressure  8 

Bacon   1 

Barnes  and  Coker   129 

Barometer   7 

Bazin's  experiments  on 

calibration  of  Pitot  tube   245 
distribution  of  pressure  in  the  plane 

of  an  orifice  59 
distribution  of  velocity  in  the  cross 

section  of  a  channel  208 
distribution  of  velocity  in  the  cross 

section  of  a  pipe   144 
distribution  of  velocity  in  the  plane  of 

an  orifice  59,  244 
flow  in  channels   182 
flow  over  dams   102 
flow  over  weirs   89 
flow  through  orifices  56 
form  of  the  jet  from  orifices  63 
Bazin's  formulae  for 
channels   182,  185 
orifices,  sharp-edged   57,  61 
velocity  at   any  depth  in   a  vertical 

section  of  a  channel  212 
velocity   at   any  point   in    the  cross 

section  of  a  pipe  144 
weir,  flat  crested  99 
weir,  sharp-crested  97-99 
weir,  sill  of  small  thickness   99 


Bends,  loss  of  head  due  to  140 
Bernouilli's  theorem   39 

applied    to    centrifugal    pumps    413, 
423,  437,  439 

applied  to  turbines   334,  349 

examples  on  48 

experimental  illustrations  of   41 

extension  of  48 
Borda's  mouthpiece   72 
Boussinesq's  theory  for  discharge  of  a 

weir   104 

Boyden  diffuser  314 
Brotherhood  hydraulic  engine  501 
Buoyancy  of  floating  bodies  21 

centre  of  23 

Canal  boats,  steering  of  47 

Capstan,  hydraulic   501 

Centre  of  buoyancy   23 

Centre  of  pressure   13 

Centrifugal  force,  effect  of  in  discharge 

from  water-wheel  286 
Centrifugal  head 

in  centrifugal  pumps  405,  408,  409, 

419,  421 

in  reaction  turbines   303,  334 
Centrifugal  pumps,  see  Pumps 
Channels 

circular,  depth  of  flow  for  maximum 

discharge  221 
circular,  depth  of  flow  for  maximum 

velocity  220 

coefficients  for,  in  formulae  of 
Bazin   186,  187 
Darcy  and  Bazin   183 
Ganguillet  and  Kutter   184 
coefficients    for,    in  logarithmic    for- 
mulae 200-208 
coefficients,  variation  of  190 
curves  of  velocity  and  discharge  for  222 
dimensions  of,  for  given  flow  deter- 
mined by  approximation   225-227 
diameter  of,  for  given  maximum  dis- 
charge 224 

distribution  of  velocity  in  cross  sec- 
tion of  208 

earth,  of  trapezoidal  form  219 
erosion  of  earth  216 


526 


INDEX 


Channels  (cont.) 

examples  on   223-231 

flow  in   178 

flow  in,  of  given  section  and  slope 

223 

forms  of 
best  218 
variety  of  178 
formula  for  flow  in 
applications  of  223 
approximate  for  earth  201,  207 
Aubisson's  233 
Bazin's  185 
Bazin's  method  of  determining  the 

constants  in   187 
Chezy  180 

Darcy  and  Bazin's  182 
Eytelwein's   181,  232 
Ganguillet  and  Kutter  s   182,  184 
historical  development  of  231 
logarithmic   192,  198-200 
Prony  181,  232 
hydraulic  mean  depth  of   179 
lined  with 

ashlar  183,  184,  186,  187,  200,  206 
boards   183,  184,  187,  195,  201 
brick  183,  184,  187,  193,  195,  197, 

203 
cement    183,    184,   186,    187,   193, 

202 

earth  183,  184,  186, 187,  201,  207 
gravel   183,  184 
pebbles  184,  186,  187,  206 
rubble  masonry  184,  186,  187,  205 
logarithmic  plottings  for   193-198 
minimum  slopes  of,  for  given  velocity 

215 

particulars  of  195 
problems   223  (see  Problems) 
sections  of  216 
siphons  forming  part  of  216 
slope  of  for  minimum  cost   227 
slopes  of  213,  215 
steady  motion  in   178 
variation  of  the  coefficient  for   190 
Coefficients 
for    mouthpieces    71,     73,    76    (see 

Mouthpieces) 

for  rectangular  notches  (see  Weirs) 
for  triangular  notches  85 
for  Venturi  meter  46 
for  weirs,   88,  89,  93  (see  Weirs) 
of  resistance  67 
Condenser   6 
Condition  of  stability  of  floating  bodies 

24 

Contraction  of  jet  from  orifice   53 
Convergent  mouthpiece   73 
Couple,  work  done  by  274 
Cranes,  hydraulic  494 
Crank  effort  diagram  'for  three  cylinder 

engine  503 
Critical  velocity  129 


Current  meters   239 

calibration  of  240 

Gurley  238 

Haskell   240 

Curved  stream  line  motion  518 
Cylindrical  mouthpiece   73 

Dams,  flow  over  101 

Darcy 

experiments  on  flow  in  channels  182 
experiments  on  flow  in  pipes   122 
formula  for  flow  in  channels   182 
formula  for  flow  in  pipes   122 

Deacon's  waste-water  meter  254 

Density  3 

of  gasoline   11 
of  kerosine   11 
of  mercury  8 
of  pure  water  4,  11 

Depth  of  centre  of  pressure   13 

Diagram  of  pressure  on  a  plane  area 
16 

Diagram  of  pressure  on  a  vertical  circle 
16 

Diagram  of  work  done  in  a  reciprocating 
pump   443,  459,  467 

Differential  accumulator  489 

Differential  gauge   8 

Discharge 

coefficient    of,    for    orifices    60    (see 

Orifices) 

coefficient  of,  for  Venturi  meter  46 
of  a  channel  178  (see  Channels) 
over  weirs  82  (see  Weirs) 
through  notches  85  (see  Notches) 
through  orifices   50  (see  Orifices) 
through  pipes   112  (see  Pipes) 

Distribution  of  velocity  on  cross  section 
of  a  channel  208 

Distribution  of  velocity  on  cross  section 
of  a  pipe   143 

Divergent  mouthpieces   73 

Dock  caisson   181,  192,  216 

Docks,  floating   31 

Drowned  nappes  of  weirs  96,  100 

Drowned  orifices  65 

Drowned  weirs  98 

Earth  channels 

approximate  formula  for   201,  207 
coefficients   for    in   Bazin's    formula 

187 
coefficients  for  in  Darcy  and  Bazin's 

formula   183 
coefficients    for    in    Ganguillet    and 

Kutter's  formula  184 
erosion  of  216 

Elbows,  loss  of  head  due  to   140 
Engines,  hydraulic   501 
Brotherhood  501 
Hastie   503 
Bigg  504 
Erosion  of  earth  channels  216 


INDEX 


527 


Examples,  solutions  to  which  are  given 

in  the  text — 
Boiler,  time   of  emptying  through  a 

mouthpiece   78 
Centrifugal  pumps,  determination  of 

pressure  head  at  inlet  and   outlet 

410 
Centrifugal  pumps,  dimensions  for  a 

given  discharge  404 
Centrifugal   pumps,    series,    number 

of  wheels  for  a  given  lift   435 
Centrifugal  pumps,  velocity  at  which 

delivery  starts  412 
Channels,    circular   diameter,    for   a 

given  maximum  discharge  224 
Channels,  diameter  of  siphon  pipes 

to    given    same    discharge    as    an 

aqueduct   224 
Channels,  dimensions  of  a  canal  for 

a  given  flow  and  slope  225,  226,  227 
Channels,     discharge     of    an    earth 

channel   225 
Channels,  flow  in,  for   given  section 

and  slope  223 
Cranes   12,  489 
Floating  docks,  height  of  metacentre 

of  34 
Floating  docks,  water  to  be  pumped 

from   33 

Head  of  water  7 
Hydraulic     machinery,     capacity    of 

accumulator    for    working    a    hy- 
draulic crane  489 
Hydraulic    motor,    variation    of    the 

pressure  on  the  plunger   470 
Impact  on   vanes,  form   of  vane  for 

water  to  enter  without  shock  and 

leave  in  a  given  direction   271 
Impact  on  vanes,  pressure  on  a  vane 

when  a  jet  in  contact  with  is  turned 

through  a  given  angle  267 
Impact     on    vanes,     turbine    wheel, 

form  of  vanes  on   272 
Impact     on    vanes,    turbine    wheel, 

water  leaving  the  vanes  of  269 
Impact   on   vanes,  work  done   on    a 

vane   271 
Metacentre,  height  of,  for  a  floating 

dock  34 

Metacentre.  height  of,  for  a  ship  26 
Mouthpiece,   discharge  through,  into 

a  condenser   76 
Mouthpiece,    time    of    emptying    a 

boiler  by  means  of  78 
Mouthpiece,    time    of    emptying    a 

reservoir  by  means  of  78 
Pipes,   diameter  of,  for  a  given  dis- 
charge  152,  153 
Pipes,  discharge  along  pipe  connecting 

two  reservoirs  151,  154 
Pipes  in  parallel   154 
Pipes,  pressure  at  end  of  a  service 

pipe  151 


Examples  (cont.) 

Pontoon,  dimensions  for  given  dis- 
placement 29 

Pressure  on  a  flap  valve   13 

Pressure  on  a  masonry  dam   13 

Pressure  on  the  end  of  a  pontoon  13 

Reciprocating  pump  fitted  with  an 
air  vessel  470 

Reciprocating  pump,  horse-power  of, 
with  long  delivery  pipe  470 

Reciprocating  pump,  pressure  in  an 
air  vessel  470 

Reciprocating  pump,  separation  in, 
diameter  of  suction  pipe  for  no  469 

Reciprocating  pump,  separation  in 
the  delivery  pipe  464 

Reciprocating  pump,  separation  in, 
number  of  strokes  at  which  sepa- 
ration takes  place  458 

Reciprocating  pump,  variation  of 
pressure  in,  due  to  inertia  forces 
470 

Reservoirs,  time  of  emptying  by  weir 
108 

Reservoirs,  time  of  emptying  through 
orifice  78 

Ship,  height  of  metacentre  of   26 

Transmission  of  fluid  pressure   12 

Turbine,  design  of  vanes  and  de- 
termination of  efficiency  of,  con- 
sidering  friction  331 

Turbine,  design  of  vanes  and  de- 
termination of  efficiency  of,  fric- 
tion neglected  322 

Turbine,  dimensions  and  form  of 
vanes  for  given  horse-power  341 

Turbine,  double  compartment  parallel 
flow  349 

Turbine,  form  of  vanes  for  an  out- 
ward flow  311 

Turbine,  hammer  blow  in  a  supply 
pipe  385 

Turbine,  velocity  of  the  wheel  for  a 
given  head  321 

Venturi  meter  46 

Water  wheel,  diameter  of  breast 
wheel  for  given  horse-power  290 

Weir,  correction  of  coefficient  for 
velocity  of  approach  94 

Weir,  discharge  of  94 

Weir,  discharge  of  by  approximation 
108 

Weir,  time  of  emptying  reservoir  by 
means  of  110 

Fall  of  free  level  51 
Fire  hose  nozzle  73 
Flap  valve,  pressure  on  18 

centre  of  18 
Floating  bodies 

Archimedes,  principle  of  22 

buoyancy  of  21 

centre  of  buoyancy  of  23 


528 


INDEX 


Floating  bodies  (cont.) 

conditions  of  equilibrium  of  21 

containing  water,  stability  of  29 

examples  on   34,  516 

metacentre  of  24 

resistance  to  the  motion  of  507 

small  displacements  of  24 

stability  of  equilibrium,  condition  of 

24 

stability  of  floating  dock  33 
stability  of  rectangular  pontoon  26 
stability  of  vessel  containing  water  29 
stability   of  vessel  wholly  immersed 

30 

weight  of  fluid  displaced  22 
Floating  docks   31 

stability  of    33 
Floats,  double  237 
rod  239 
surface  237 
Flow  of  water 

definitions  relating  to   38 
energy  per  pound  of  flowing  water  38 
in  open  channels  178  (see  Channels) 
over  dams   101  (see  Dams) 
over  weirs  81  (see  Weirs) 
through  notches   80  (see  Notches) 
through  orifices  50  (see  Orifices) 
through  pipes  112  (see  Pipes) 
Fluids  (liquids) 
at  rest  3-19 
examples  on   19 
compressible   3 
density  of  3 

flow  of,  through  orifices  50 
incompressible  3 
in  motion   37 

pressure  in,  is  the  same  in  all  direc- 
tions 4 

pressure  on  an  area  in  12 
pressure  on  a  horizontal  plane  in,  is 

constant  5 
specific  gravity  of  3 
steady  motion  of  37 
stream  line  motion  in  37,  517 
transmission  of  pressure  by   11 
used  in  U  tubes  9 
viscosity  of  2 

Forging  press,  hydraulic   492 
Fourneyron  turbine  307 
Friction 

coefficients  of,  for  ships'  surfaces  509, 

515 
effect  of,  on  discharge  of  centrifugal 

pump   421 

effect  of,  on  velocity  of  exit  from  Im- 
pulse Turbine  373 
effect   of,  on   velocity    of  exit   from 

Poncelet  Wheel  297 
Froude's  experiments  on  fluid  507 
in  centrifugal  pumps  400 
in  channels  180 
in  pipes  113,  118 


Friction  (cont.) 

in  reciprocating  pumps  449 
in  turbines  313,  321,  339,  373 

Ganguillet  and  Kutter 

coefficients  in  formula  of  125,  184 

experiments  of  183 

formula  for  channels   184 

formula  for  pipes  124 
Gasoline,  specific  gravity  of  11 
Gauges,  pressure 

differential  8 

inverted  air   9 

inverted  oil   10 
Gauging  the  flow  of  water   234 

by  an  orifice  235 

by  a  weir   247 

by  chemical  means  258 

by  floats  239  (see  Floats) 

by  meters  234,  251  (see  Meters) 

by  Pitot  tubes  241 

by  weighing   234 

examples  on   260 

in  open  channels  236 

in  pipes   251 

Glazed  earthenware  pipes   186 
Gurley's  current  meter   238 

Hammer  blow  in  a  long  pipe  384 
Haskell's  current  meter   240 
Hastie's  engine   503 
Head 

position   39 

pressure   7,  39 

velocity  39 

High  pressure  pump  471 
Historical    development    of    pipe    and 

channel  formulae   231 
Hook  gauge   248 
Hydraulic  accumulator    486 
Hydraulic  capstan   501 
Hydraulic  crane   494 

double  power  495 

valves  497 

Hydraulic  differential  accumulator  490 
Hydraulic  engines   501 

crank  effort  diagram  for  503 
Hydraulic  forging  press  492 
Hydraulic  gradient   115 
Hydraulic  intensifier  491 
Hydraulic  machines  485 

conditions     which    vanes    of,     must 
satisfy  270 

examples  on   489,  505 

joints  for  485 

maximum  efficiency  of  295 

packings  for  485 
Hydraulic  mean  depth  119 
Hydraulic  motors,  variations  of  pressure 

in,  due  to  inertia  forces  469 
Hydraulic  ram  474 
Hydraulic  riveter   499 
Hydraulics,  definition  of  1 


INDEX 


529 


Hydrostatics  4-19 

Impact  of  water  on  vanes  261  (see  Vanes) 
Inertia  forces  in  hydraulic  motors  469 
Inertia  forces  in  reciprocating  pumps 

44o 

Inertia,  moment  of  14 
Inverted  air  gauge  9 
Inverted  oil  gauge   9 
Intensifies,  hydraulic   491 

non-return  valves  for  492 
Intensitiers,  steam  493 
Inward    flow    turbines    275,    318    (see 
Turbines) 

Joints  used  in  hydraulic  work   485 

Kennedy  meter  255 
Kent  Venturi  meter  253 
Kerosene,  specific  gravity  of  11 

Leathers  for  hydraulic  packings  486 
Logarithmic  formulae  for  flow 

in  channels   192 

in  pipes  125 
Logarithmic  plottings 

for  channels  195 

for  pipes  127,  133 
Luthe  valve  499 

Masonry  dam   17 
Mercury 

specific  gravity  of  8 

use  of,  in  barometer   7 

use  of,  in  U  tubes   8 
Metacentre,  height  of  24 
Meters 

current   239 

Deacon's  waste  water  254 

Kennedy  255 

Leinert  234 

Venturi   44,  75,  251 
Moment  of  inertia  14 

of  water  plane  of  floating  body  25 

table  of   15 

Motion,  second  law  of  263 
Mouthpieces  54 

Borda's   72 

coefficients  of  discharge  for 
Borda's   73 
conical  73 
cylindrical   71,  76 
fire  nozzle  73 

coefficients  of  velocity  for   71,  73 

conical  73 

convergent   73 

cylindrical   73 

divergent   73 

examples  on   78 

flow  through,  under  constant  pressure 
75 

loss  of  head  at  entrance  to   70 

time  of  emptying  boiler  through   78 

L.  H. 


Mouthpieces  (cont.) 

time   of  emptying  reservoir  through 

78 

Nappe  of  a  weir  81 

adhering  95 

depressed  95 

drowned  or  wetted   95 

free  95 

instability  of  the  form  of  97 
Newton's  second  law  of  motion   263 
Notation  used  in  connection  with  vanes, 
turbines  and  centrifugal  pumps  272 
Notches 

coefficients  for  rectangular  (see  Weirs) 

coefficients  for  triangular  85 

rectangular  80  (see  Weirs) 

triangular  80 

Nozzle  at  end  of  a  pipe  159 
Nozzle,  fire  74 

Oil  pressure  gauge,  inverted  10 

calibration   of  11 

Oil  pressure  regulator  for  turbines  377 
Orifices 

Bazin's  coefficients  for  57,  61 
Bazin's  experiments  on  56 
coefficients  of  contraction  52,  56 
coefficients  of  discharge   57,   60,  61, 

63 

coefficients  of  velocity   54,  57 
contraction  complete  53,  57 
contraction  incomplete  or  suppressed 

53,  63 
distribution   of  pressure  in  plane  of 

59 

distribution  of  velocity  in  plane  of  59 
drowned  65 
drowned  partially  66 
examples  on   78 
flow  of  fluids  through   50 
flow  of  fluids  through,  under  constant 

pressure  75 
force  acting  on  a  vessel  when  water 

issues  from  277 
form  of  jet  from   63 
large  rectangular   64 
partially  drowned  66 
pressure  in  the  plane  of  59 
sharp-edged  52 
time  of  emptying  a  lock  or  tank  by 

76,  77 

Torricelli's  theorem  51 
velocity  of  approach  to  66 
velocity  of  approach  to,  effect  on  dis- 
charge from  67 

Packings  for  hydraulic  machines  485 
Parallel  flow  turbine  276,  342,  368 
Parallel  flow  turbine  pump   437 
Pelton  wheel  276,  377,  380 
Piezometer  fittings   139 
Piezometer  tubes   7 

34 


530 


INDEX 


Pipes 

bends,  loss  of  head  due  to  141 
coefficients 

C  in  formula    v  =  G*Jmi, 


and 


/-HB 


for  cast  iron,  new  and  old  120, 

121,  122,  123,  124 
for  steel  riveted   121 
for  Darcy's  formula   122 
for  logarithmic  formulae 
brass  pipes   133,  138 
cast  iron,  new  and  old  125,  137, 

138 

glass   135 
riveted   137,  138 
wood  135,  138 
wrought  iron   122,  135,  138 
n  in  Ganguillet  and  Kutter's  formula 
cast  iron,  new  and  old  125 
for  glazed  earthenware  125 
for  steel  riveted  184 
for  wood  pipes  125,  184 
variation  of,  with  service  123 
connecting  three  reservoirs   155 
connecting  two  reservoirs   149 
connecting  two  reservoirs,  diameter  of 

for  given  discharge   152 
critical  velocity  in   128 
Darcy's  formula  for   122 
determination   of    the   coefficient   C, 
as  given  in  tables   by  logarithmic 
plotting  132 
diameter     of,    for     given    discharge 

152 

diameter  for  minimum  cost   158 
diameter  varying   160 
divided  into  two  branches   154 
elbows  for   141 

empirical  formula  for  head  lost  in  119 
empirical  formula  for  velocity  of  flow 

in   119 

equation  of  flow  in   117 
examples  on  flow  in  149-162,  170 
experimental  determination  of  loss  of 

head  by  friction  in   116 
experiments  on  distribution  of  velocity 

experiments  on  flow  in,  criticism  of 

138 
experiments  on  loss  of  head  at  bends 

142 
experiments  on  loss  of  head  in   122 

125,  129,  131,  132,  136 
experiments    on    loss    of    head    in, 

criticism  of  138 
flow  through    112 
flow  diminishing  at  uniform  rate  in 

157 
formula  for 

Chezy   119 

Darcy   122 


Pipes  (cont.) 

formula  for  (cont.) 

logarithmic  125,  131,  133,  137-138 

Reynolds   131 

summary  of  148 

velocity   at  any  point  in  a  cross 

section  of   143 
friction  in,  loss  of  head  by  113 

determination  of  116 
Ganguillet  and  Kutter's  formula  for 

124 

gauging  the  flow  in  251 
hammer  blow  in   384 
head  lost  at  entrance  of  70,  114 
head  lost  by  friction  in   113 
head   lost    by  friction    in,   empirical 

formula  for   119 
head  lost  by  friction  in,  examples  on 

150-162,  170 
head  lost  by  friction  in,  logarithmic 

formula  for   125,  133 
head    required    to    give    velocity    to 

water  in  the  pipe  146 
head  required  to  give  velocity  to  water 

in  the  pipe,  approximate  value   113 
hydraulic  gradient  for  115 
hydraulic  mean  depth  of  118 
joints  for  485 
law  of  frictional  resistance  for,  above 

the  critical  velocity   130 
law  of  frictional  resistance  for,  below 

the  critical  velocity   125 
limiting  diameter  of  165 
logarithmic  formula  for  125 
logarithmic  formula  for,   coefficients 

in  138 

logarithmic  formula,  use  of,  for  prac- 
tical calculations   136 
logarithmic  plottings  for   126 
nozzle  at  discharge  end  of,  area  of 

when  energy  of  jet  is  a  maximum 
159 

when  momentum  of  jet  is  a  maxi- 
mum 159 

piezometer  fittings  for   139 
pressure  on  bends  of  166 
pressure  on  a  cylinder  in   169 
pressure  on  a  plate  in   168 
problems   147  (see  Problems) 
pumping   water   through    long   pipe, 

diameter  of  for  minimum  cost  158 
resistance  to  motion  of  fluid  in   112 
rising  above  hydraulic  gradient   115 
short   153 
siphon   161 
temperature,  effect  of,  on  velocity  of 

flow  in    131,  140 

transmission  of  power  along,  by  hy- 
draulic pressure  162 
values  of  C  in  the  formula  v  =  C  \lrni 

for   120,  121 
variation  of  C  in  the  formula  v  = 

for   123 


INDEX 


531 


Pipes  (cont.) 

variation  of   the   discharge  of,   with 

service   123 

velocity  of  flow  allowable  in   162 
velocity,  head  required  to  give  velocity 

to  water  in    146 
velocity,  variation  of,  in  a  cross  section 

of  a  pipe  143 
virtual  slope  of  115 
Pitot  tube   241 

calibration  of  245 
Poncelet  water  wheel  294 
Pontoon,  pressure  on  end  of  18 
Position  head   29 
Press,  forging   493 
Press,  hydraulic  493,  498 
Pressure 

at  any  point  in  a  fluid   4 
atmospheric,  in  feet  of  water   8 
gauges   8 
head   7 

measured  in  feet  of  water   7 
on  a  horizontal  plane  in  a  fluid   5 
on  a  plate  in  a  pipe   168 
on  pipe  bends   166 
Principle  of  Archimedes   19 
Principle  of  similarity   84 
Problems,  solutions  of  which  are  given 

in  the  text — 
channels 

diameter  of,  for  a  given  maximum 

discharge  224 
dimensions    of,    for    a    given   flow 

225-227 

earth  discharge  along,  of  given  di- 
mensions and  slope   224 
flow  in,  of  given  section  and  slope 

223 

slope  of,  for  minimum  cost   227 
solutions    of,     by     approximation 

225-227 
pipes 

acting  as  a  siphon   161 
connecting  three  reservoirs   155 
connecting  two  reservoirs    149 
diameter  of,  for  a  given  discharge 

152 

divided  into  two  branches   154 
head  lost  in,  when  flow  diminishes 

at  uniform  rate   157 
loss  of  head  in,  of  varying  diameter, 

160,  161 
pumping  water  along,  diameter  of, 

for  minimum  cost   158 
with  nozzle  at  the  end    158,  159 
Propulsion  of  ships  by  water  jets    279 
Pumping  water  through  long  pipes  158 
Pumps 

centrifugal   392 
advantages  of   435 
Appold  415 

Bernoulli's    equation    applied    to 
413 


Pumps  (cont.) 
centrifugal  (cont.) 
centrifugal  head,  effect  of  variation 

of  on  discharge  421 
centrifugal  head,  impressed  on  the 

water  by  the  wheel   405 
design  of,  for  given  discharge  402 
discharge,    effect  of    the  variation 

of  tne  centrifugal  head  and  loss 

by  friction  on  419 
discharge,    head-velocity    curve    at 

zero  409 
discharge,    variation    of    with    the 

head  at  constant  speed  410 
discharge,  variation  of  with  speed 

at  constant  head  410 
efficiencies  of  401 
efficiencies     of,     experimental    de- 
termination of  401 
examples   on   404,   412,   414,    418, 

435,  478 

form  of  vanes  396 
friction,  effect  of  on  discharge  419, 

421 
general  equation  for  421,  425,  428, 

430 

gross  lift  of  400 
head-discharge   curve    at   constant 

velocity  410,  412,  427 
head  lost  in  414 
head,  variation   of  with  discharge 

and  speed   418 
head-velocity    curve    at    constant 

discharge  429 

head-velocity    curve    at    zero    dis- 
charge 409 
kinetic    energy    of    water    at   exit 

399 
limiting    height    to    which    single 

wheel  pump  will  raise  water  431 
limiting  velocity  of  wheel   404 
losses  of  head  in   414 
multi-stage  433 
series  433 

spiral  casing  for  394,  429 
starting  of  395 
suction  of   431 
Sulzer  series  434 
Thomson's  vortex  chamber  397,  407, 

422 
triangles  of  velocities  at  inlet  and 

exit   397 
vane  angle  at  exit,  effect  of  variation 

of  on  the  efficiency  415 
velocity-discharge  curve  at  constant 

head  411,  412,  421,  428 
velocity,  head-discharge  curve  for  at 

constant  410 
velocity  head,  special  arrangement 

for  converting  into  pressure  head 

422 
velocity,  limiting,  of  rim  of  wheel 

404 


532 


INDEX 


Pumps  (cont.) 
centrifugal  (cont.) 
velocity  of  whirl,  ratio  of,  to  velocity 

of  outlet  edge  of  vane  398 
vortex  chamber  of  397,  407,  422 
with  whirlpool  or  vortex  chamber 

397,  407,  422 

work  done  on  water  by   397 
compressed  air  477 
duplex  473 

examples  on  458,  464,  469,  478 
force  392 
high  pressure  472 
hydraulic  ram  476 
packings  for  plungers  of  472,  486 
reciprocating  439 

acceleration,  effect   of  on  pressure 

in  cylinder  of  a  446,  448 
acceleration  of  the  plunger  of  444 
acceleration  of  the  water  in  delivery 

pipe  of  448 
acceleration  of  the  water  in  suction 

pipe  of  445 

air  vessel  on  delivery  pipe  of  454 
air  vessel  on  suction  pipe  of  451 
air  vessel  on  suction  pipe,  effect  of 

on  separation   462 
coefficient  of  discharge  of  442 
diagram  of  work  done  by  443,  450, 

459,  467 

discharge,  coefficient  of  443 
duplex  473 
examples    on    458,   464,  469,  470, 

480 
friction,  variation  of  pressure  in  the 

cylinder  due  to  449 
head  lost  at  suction,  valve  of  468 
head  lost  by  friction  in  the  suction 

and  delivery  pipes  449 
high  pressure  plunger  471 
pressure  in  cylinder  of  when  the 

plunger  moves  with  simple  har- 
monic motion  446 
pressure  in  the  cylinder,  variation 

of  due  to  friction  449 
separation  in  delivery  pipe  463 
separation    during    suction    stroke 

456 
separation    during    suction    stroke 

when  plunger  moves  with  simple 

harmonic  motion  458,  461 
slip  of  442,  461 
suction  stroke  of    441 
suction  stroke,  separation  in   456, 

461,  462 

Tangye  duplex  473 
vertical  single  acting  440 
work  done  by  441 
work  done  by,  diagram  of  443,  459, 

turbine  396,  425 

head-discharge  curves  at   constant 
speed  427 


Pumps  (cont.) 
turbine  (cont.) 
head-velocity    curves    at    constant 

discharge  429 
inward  flow  439 
multi-stage  433 
parallel  flow   437 
velocity-discharge  curves  at  constant 

head  428 
Worthington    432 
work  done  by  443 
work  done  by,  diagram   of    (see  Re- 

ciprocating  pumps) 
work  done  by,  series   433 

Eeaction  turbines   301 

limiting  head  for  367 

series   367 
Reaction  wheels   301 

efficiency  of  304 

Reciprocating  pumps   439  (see   Pumps) 
Rectangular  pontoon,  stability  of   26 
Rectangular  sharp-edged  weir  81 
Rectangular  sluices   65 
Rectangular    weir    with    end    contrac- 
tions 88 
Regulation  of  turbines    306,  317,  318, 

323,  348 
Regulators 

oil  pressure,  for  impulse  turbine  377 

water   pressure,  for  impulse  turbine 

379 
Relative  velocity  265 

as  a  vector  266 
Reservoirs,  time   of  emptying   through 

orifice   76 
Reservoirs,  time  of  emptying  over  weir 

109 

Resistance  of  ship   510 
Rigg  hydraulic  engine   503 
Rivers,  flow  of   191,  207,  211 
Rivers,  scouring  banks  of  520 
Riveter,  hydraulic  500 

Scotch  turbine  301 

Second  law  of  motion   263 

Separation  (see  Pumps) 

Sharp-edged  orifices 
Bazin's  experiments  on   56 
distribution  of  velocity  in  the  plane  of  59 
pressure  in  the  plane  of  59 
table  of  coefficients   for,    when    con- 
traction is  complete   57,  61 
table  of  coefficients   for,  when    con- 
traction is  suppressed   63 

Sharp-edged  weir  81  (see  Weirs) 

Ships 

propulsion  of  by  water  jets  279 
resistance  of  510 
resistance  of,  from  model   515 
stream  line  theory  of  the  resistance 
of  510 

Similarity,  principle  of  84 


INDEX 


533 


"Siphon,  forming  part  of  aqueduct   216 

pipe   161 

Slip  of  pumps  442,  461 
Sluices   65 

for  regulating  turbines  (see  Turbines) 
Specific  gravity  3 

of  gasoline   11 

of  kerosene   11 

of  mercury   8 

of  oils,  variation  of,  with  temperature 
11 

of  pure  water   4 

variation  of,  with  temperature   11 
"Stability  of 

floating  body  24,  25 

floating  dock  31 

floating  vessel  containing  water  29 

rectangular  pontoon   26 
Steady  motion  of  fluids   37 
Steam  intensifier  493 
Stream  line  motion   37,  129,  517 

curved  518 

Hele  Shaw's  experiments  on  284 
Stream    line    theory    of   resistance    of 

ships  510 

Suction  in  centrifugal  pump  431 
Suction  in  reciprocating  pump  441 
Suction  tube  of  turbine   306 
Sudden    contraction    of    a    current    of 

water  69 
Sudden   enlargement   of  a    current   of 

water  67 

Sulzer,  multi-stage  pump  434 
Suppressed  contraction   53 

effect   of,  on    discharge   from  orifice 
62 

effect  of,  on  discharge  of  a  weir  82 

Tables 

channels,  sewers  and  aqueducts,  par- 

ticulars   of,    and   values    of    -  in 

kvn  P 

formula   i= —      195 
mP 

channels 

slopes  and  maximum  velocities  of 

flow  in   215 
values  of  a  and  ft  in  Bazin's  formula 

183 

values  of  v  and  i  as  determined 
experimentally  and  as  calculated 
from  logarithmic  formulae  198, 
201-208 

coefficients  for  dams  102 
coefficients    for    sharp-edged    orifice, 

contraction  complete  57,  61 
coefficients    for    sharp-edged    orifice, 

contraction  suppressed  63 
coefficients  for  sharp-edged  weirs  89, 

93 

coefficients  for  Venturi  meters  46 
earth  channels,  velocities  above  which 
erosion  takes  place   216 


Tables  (cow*.) 

minimum  slopes  for  varying  values 
of  the  hydraulic  mean  depth  of 
brick  channels  that  the  velocity 
may  not  be  less  than  2  ft.  per 
second  215 

moments  of  Inertia  15 

Pelton  wheels,  particulars  of  377 

pipes 

lead,  slope  of  and  velocity  of  flow 

in   128 
reasonable  values  of  7  and   n  in 

the  formula  h  =  ?S-   138 


values     of_   C     in     the     formula 

v  =  C»Jmi   120,  121 
values     of     /     in     the      formula 


values    of    n    in    Ganguillet    and 

Kutter's  formula  125,  184 
values  of  n  and  k  in  the  formula 

i  =  kvn  137 
resistance  to   motion   of    boards    in 

fluids   509 

turbines,  peripheral  velocities  and 
heads  of  inward  and  outward  flow 
333 

useful  data  3 
Thomson,     centrifugal     pump,     vortex 

chamber  for  397,  407,  422 
principle  of  similarity   62 
turbine  323 
Time  of  emptying  tank  or  reservoir  by 

an  orifice    76 
Time  of  emptying  a  tank  or  reservoir 

by  a  weir   109 
Torricelli's  theorem   1 

proof  of  51 
Total  pressure   12 
Triangular  notches  80 

discharge  through   85 
Turbines 

axial  flow  276,  342 

axial  flow,  impulse  368 

axial  flow,  pressure  or  reaction  342 

axial  flow,  section  of  the  vane  with 

the  variation  of  the  radius  344 
Bernouilli's  equations  for   334 
best  peripheral  velocity  for  329 
central  vent  320 
centrifugal  head  impressed  on  water 

by  wheel  of  334 
cone   359 

design  of  vanes  for  346 
efficiency  of  315,  331 
examples  on   311,  321,  323,  331,  341, 

349,  385,  387 

flow  through,  effect  of  diminishing, 
by  means  of  moveable  guide  blades 
362 

flow  through,  effect  of  diminishing 
by  means  of  sluices  364 


534 


INDEX 


Turbines  (cont.) 
flow   through,   effect   of   diminishing 

on  velocity  of  exit   363 
Fontaine,  regulating  sluices  348 
form  of  vanes  for  308,  347,  365 
Fourneyron  306 
general  formula  for  31 
general    formula,    including   friction 

315 
guide  blades  for   320,  326,  348,  352, 

362 
guide  blades,  effect   of  changing  the 

direction  of  362 
guide  blades,  variation  of  the   angle 

of,  for  parallel  flow  turbines   344 
horse    power,    to    develop     a    given 

339 
impulse  300,  369-384 

axial  flow   368 

examples  387 

for  high  heads  373 

form  of  vanes  for  371 

Girard  369,  370,  373 

hydraulic  efficiency  of  371,  373 

in  airtight  chamber  370 

oil  pressure  regulator  for  377 

radial  flow  370 

triangles  of  velocities   for   372 

triangles  of  velocities  for  considering 
friction  373,  376 

water  pressure  regulator  for  379 

water  pressure  regulator,  hydraulic 
valve  for  382 

water  pressure  regulator,  water  filter 
for  383 

work  done  on  wheel  per  Ib.  of  water 

272,  277,  323 
inclination  of  vanes  at  inlet  of  wheel 

308,  321,  344 
inclination  of  vanes  at  outlet  of  wheel 

308,  321,  345 
in  open  stream   360 
inward  flow   275,  318 

Bernouilli's  equations  for  334,  339 

best    peripheral     velocity    for,     at 
inlet  329 

central  vent  320 

examples  on  321,  331,  341,  387 

experimental  determination  of  the 
best  velocity  for  329 

for  low  and  variable  falls  328 

Francis  320 

horizontal  axis  327 

losses  in   321 

Thomson   324 

to    develop    a    given    horse- power 
339 

triangles  of  velocities  for  322,  326, 
332 

work  done  on  the  wheel  per  Ib.  of 

water  321 
limiting    head    for    reaction    turbine 

367 


Turbines  (cont.) 

loss  of  head  in  313,  321 
mixed  flow  350 

form  of  vanes  of  355 

guide    blade    regulating    gear    for 
352-354 

in  open  stream  360 

Swain  gate  for   374 

triangles     of    velocities    for     355- 
356 

wheel  of  351 
Niagara  falls   318 
oil  pressure  regulator  for   377 
outward  flow,  275,  306 

Bernouilli's     equations     for     334r 
339 

best  peripheral  velocity  for,  at  inlet 
329 

Boyden   314 
diffuser  for  314 

double   316 

examples  on  311,  387 

experimental  determination  of  the 
best  velocity   for  329 

Fourneyron  307 

losses  of  head  in   313 

Niagara  falls   318 

suction  tube  of   308,  317 

triangles  of  velocities  for   308 

work  done  on  the  wheel  per  Ib.  of 

water  310,  315 
parallel  flow  276,  342 

adjustable  guide  blades  for   348 

Bernouilli's  equations  for  348 

design  of  vanes    for  344 

double  compartment  343 

examples  on  349,  387 

regulation  of  the  flow  to   348 

triangles  of  velocities  for   344 
reaction  301 

axial  flow   276-342 

cone   359 

inward  flow  275,  318 

mixed  flow  350 

outward  flow  306 

parallel  flow   276-342 

Scotch   302 

series  368 
regulation  of  306,  317,  318,  323,  348, 

350,  352,  360,  362,  364 
Scotch  301 
sluices  for   305,  307,    316,   317,  319, 

327,  328,  348,  350,  361,  364 
suction  tube  of   306 
Swain  gate  for  364 
Thomson's  inward  flow   323 
to  develop  given  horse- power  339 
triangles   of    velocities    at   inlet   and 

outlet  of  impulse  372,  376 
triangles    of   velocities   at   inlet  and 

outlet  of  inward  flow  308 
triangles   of  velocities    at   inlet   and 

outlet  of  mixed  flow   356 


INDEX 


535 


Turbines  (cont.) 

triangles   of   velocities    at   inlet   and 

outlet  of  outward  flow  344 
triangles    of   velocities   at    inlet   and 

outlet  of  parallel  flow   344 
types  of  300 
vanes,  form  of 

between  inlet  and  outlet  365 

for  inward  flow  321 

for  mixed  flow  351,  356 

for  outward  flow   311 

for  parallel  flow   344 
velocity  of  whirl   273,  310 

ratio  of,    to  velocity  of  inlet  edge 

of  vane  332 

velocity  with  which  water  leaves  334 
wheels,  path  of  water  through   312 
wheels,  peripheral  velocity  of  333 
Whitelaw  302 
work  done  on  per  Ib.  of   flow,    275, 

304,  315 

Turning  moment,  work  done  by  273 
Tweddell's  differential  accumulator  489 

U  tubes,  fluids  used  in  9 
Undershot  water  wheels   292 

Valves 

crane  497 

hydraulic  ram   476 

intensiner  492 

Luthe  499 

pump  470-472 
Vanes 

conditions  which  vanes  of  hydraulic 
machines  should  satisfy   270 

examples  on  impact  on   269,  272,  280 

impulse  of  water  on   263 

notation    used    in    connection    with 
272 

Pelton  wheel  276 

pressure  on  moving  266 

work  done  266,  271,  272,  275 
Vectors 

definition  of  261 

difference  of  two   262 

relative    velocity    defined    as    vector 
266 

sum  of  two  262 
Velocities,  resultant  of  two   26 
Velocity 

coefficient  of,  for  orifices    54 

head  39 

of  approach  to  orifices   66 

of  approach  to  weirs  90 

relative  265 

Venturi  meter  44,  75,  251 
Virtual  slope   115 
Viscosity  2 

Water 

definitions  relating  to  flow  of  38 


Water  (cant.) 
density  of  3 
specific  gravity  of  3 
viscosity  of  2 
Water  wheels 
Breast  288 
effect  of  centrifugal  forces  on  water 

286 

examples  on   290,  386 
Impulse  291 
Overshot   283 
Poncelet  294 
Sagebien   290 

Undershot,  with  flat  blades  292 
Weirs 

Bazin's  experiments  on  89 
Boussinesq's  theory  of  104 
coefficients 

Bazin's  formula  for 
adhering  nappe  98 
depressed  nappe   98 
drowned  nappe  97 
flat  crested  99,  100 
free  nappe  88,  98 

Bazin's  tables  of  89,  93 

for  flat-crested  99,  100 

for  sharp-crested  88,  89,  93,  97, 
98 

for  sharp-crested,  curve  of  90 

Rafter's  table  of   89 
Cornell  experiments  on   89 
dams  acting  as,  flow  over   101 
discharge  of,   by  principle   of    simi- 
larity 86 

discharge   of,    when    air   is    not  ad- 
mitted below  the  nappe  94 
drowned,  with  sharp  crests   98 
examples  on   93,  98,  108,  110 
experiments  at  Cornell   89 
experiments  of  Bazin   89 
flat-crested   100 

form  of,  for  accurate  gauging   104 
formula  for,  derived  from  that  of  a 

large  orifice   82 
Francis'  formula  for    83 
gauging  flow  of  water  by  247 
nappe  of 

adhering  9j,  96 

depressed   95,  98,  99 

drowned  95,  96,  98 

free   88,  95,  98 

instability  of  97 

wetted  95,  96,  99 
of  various  forms   101 
principle  of  similarity  applied  to   86 
rectangular  sharp-edged  81 
rectangular,    with    end    contractions 

82 

side  contraction,  suppression  of  82 
sill,   influence  of  the  height  of,   on 

discharge  94 
sill  of  small  thickness  99 


536 


INDEX 


Weirs  (cont.) 

time    required    to    lower    water    in 

reservoir  by  means  of  109 
various  forms  of  101 
velocity    of  approach,    correction    of 

coefficient  for  92 
velocity   of    approach,    correction   of 

coefficient  for,  examples  on   94 


Weirs  (cont.) 

velocity   of    approach,    effect    of    on 
discharge  90 

wide  flat-crested   100 
Whitelaw  turbine   302 
Whole  pressure   12 
Worthington  multi-stage  pump   433 


CAMBRIDGE:  PRINTED  BY  JOHN  CLAY,  M.A.  AT  THE  UNIVERSITY  PRESS. 


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OCT  21  1940 

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